Difference between revisions of "Relative Extrema and Convex Functions"
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| − | + | ==Convex and Concave Functions== | |
| + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | ||
| + | :'''Definition''': A function <math>f : I \to \mathbb{R}</math> is said to be '''Convex''' if for every <math>x, y \in I</math> and for every <math>t \in [0, 1]</math> we have that | ||
| + | :<math>f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y)</math>. | ||
| + | :A function <math>f : I \to \mathbb{R}</math> is said to be '''Concave''' if for every <math>x, y \in I</math> and for every <math>t \in [0, 1]</math> we have that | ||
| + | :<math>f(tx + (1 - t)y) \geq tf(x) + (1 - t)f(y)</math>.</td> | ||
| + | </blockquote> | ||
| − | http://mathonline.wikidot.com/convex-and-concave-functions | + | We now give equivalent definitions for convex and concave functions. |
| + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | ||
| + | :'''Theorem 1:''' Let <math>f : I \to \mathbb{R}</math>.<br /> | ||
| + | :'''a)''' <math>f</math> is convex on <math>I</math> if and only if for all <math>a, b, c \in I</math> with <math>a < b < c</math> we have that <math>\displaystyle{\frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(a)}{c - a}}</math>.<br /> | ||
| + | :'''b)''' <math>f</math> is concave on <math>I</math> if and only if for all <math>a, b, c \in I</math> with <math>a < b < c</math> we have that <math>\displaystyle{\frac{f(b) - f(a)}{b - a} \geq \frac{f(c) - f(a)}{c - a}}</math>.</td> | ||
| + | </blockquote> | ||
| + | |||
| + | We only prove (a) above. The proof of (b) is analogous.</em></p> | ||
| + | *'''Proof of a):''' Let ><math>a, b, c \in I</math> be such that <math>a < b < c</math>.</li> | ||
| + | |||
| + | *<math>\Rightarrow</math> Suppose that <math>f</math> is convex on <math>I</math>. Then for all <math>t \in [0, 1]</math> we have that:</li> | ||
| + | <div style="text-align: center;"><math>\begin{align} \quad f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y) \end{align}</math></div> | ||
| + | *Set <math>a = x</math>, <math>b = tx + (1-t)y</math>, and <math>c = y</math>. Combining the first and third equations with the second equation gives us:</li> | ||
| + | <div style="text-align: center;"><math>\begin{align} \quad b = ta + (1-t)c \end{align}</math></div> | ||
| + | *Solving for <math>t</math> gives us:</li> | ||
| + | <div style="text-align: center;"><math>\begin{align} \quad b = ta + c - tc \\ \quad b - c = t(a - c) \\ \end{align}</math></div> | ||
| + | *Therefore: | ||
| + | <div style="text-align: center;"><math>\begin{align} \quad t = \frac{b - c}{a - c} = \frac{c - b}{c - a} \end{align}</math></div> | ||
| + | *Similarly, we compute <math>1 - t</math> to be:</li> | ||
| + | <div style="text-align: center;"><math>\begin{align} \quad 1 - t = \frac{c - a}{c - a} - \frac{c - b}{c - a} = \frac{b - a}{c - a} \end{align}</math></div> | ||
| + | *From the convexity of <math>f</math> we have <math>f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)</math>, or equivalently:</li> | ||
| + | <div style="text-align: center;"><math>\begin{align} \quad f(b) \leq \frac{c - b}{c - a}f(a) + \frac{b - a}{c - a}f(c) \\ \end{align}</math></div> | ||
| + | *And hence:</li> | ||
| + | <div style="text-align: center;"><math>\begin{align} \quad (c - a)f(b) \leq (c - b)f(a) + (b - a)f(c) = (c - a + a - b)f(a) + (b - a)f(c) = (c - a)f(a) - (b - a)f(a) + (b - a)f(c) \end{align}</math></div> | ||
| + | *Therefore:</li> | ||
| + | <div style="text-align: center;"><math>\begin{align} \quad (c - a)[f(b) - f(a)] \leq (b - a)[f(c) - f(a)] \quad \Leftrightarrow \quad \frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(a)}{c - a} \end{align}</math></div> | ||
| + | *<math>\Leftarrow</math> Obtained by working backwards from above. <math>\blacksquare</math></li> | ||
| + | |||
| + | We state yet another important definition for convex and concave functions.</p> | ||
| + | |||
| + | <blockquote style="background: white; border: 1px solid black; padding: 1em;"> | ||
| + | :'''Theorem 2:''' Let ><math>f : I \to \mathbb{R}</math>.<br /> | ||
| + | :'''a)''' <math>f</math> is convex on ><math>I</math> if and only if for all ><math>a, b, c \in I</math> with ><math>a <; b <; c</math> we have that ><math>\displaystyle{\frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(b)}{c - b}}</math>.<br /> | ||
| + | :'''b)''' <math>f</math> is concave on ><math>I</math> if and only if for all ><math>a, b, c \in I</math> with ><math>a <; b <; c</math> we have that ><math>\displaystyle{\frac{f(b) - f(a)}{b - a} \geq \frac{f(c) - f(b)}{c - b}}</math>.</td> | ||
| + | </blockquote> | ||
| + | |||
| + | Theorem 2 gives us a nice characterization of convex functions. It tells us that a function <math>f : I \to \mathbb{R}</math> is convex if and only if whenever we take three points <math>a, b, c \in I</math> with <math>a <; b <; c</math> we have that the slope of the line connecting <math>(a, f(a))</math> and <math>(b, f(b))</math> is less than or equal to the sope of the line connecting <math>(b, f(b))</math> and <math>(c, f(c))</math>. In other words, the slope of the line segments connecting consecutive pairs of points on the graph of <math>f</math> is increasing.</p> | ||
| + | |||
| + | We can combine theorems 1 and 2 to get a nice chain of inequalities. That is, <math>f : I \to \mathbb{R}</math> is convex if and only if for all <math>a, b, c \in I</math> with ><math>a < b < c</math> we have that:</p> | ||
| + | <div style="text-align: center;"><math>\begin{align} \frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(a)}{c - a} \leq \frac{f(c) - f(b)}{c - b} \end{align}</math></div> | ||
| + | |||
| + | ==Licensing== | ||
| + | Content obtained and/or adapted from: | ||
| + | * [https://en.wikipedia.org/wiki/Maxima_and_minima] under a CC BY-SA license | ||
| + | * [http://mathonline.wikidot.com/convex-and-concave-functions] under a CC BY-SA license | ||
Revision as of 17:22, 22 October 2021
Convex and Concave Functions
- Definition: A function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : I \to \mathbb{R}} is said to be Convex if for every and for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \in [0, 1]} we have that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y)} .
- A function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : I \to \mathbb{R}} is said to be Concave if for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x, y \in I} and for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \in [0, 1]} we have that
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(tx + (1 - t)y) \geq tf(x) + (1 - t)f(y)} .
We now give equivalent definitions for convex and concave functions.
- Theorem 1: Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : I \to \mathbb{R}} .
- a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is convex on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} if and only if for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in I} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a < b < c} we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(a)}{c - a}}} .
- b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is concave on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} if and only if for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in I} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a < b < c} we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\frac{f(b) - f(a)}{b - a} \geq \frac{f(c) - f(a)}{c - a}}} .
We only prove (a) above. The proof of (b) is analogous.
- Proof of a): Let >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in I} be such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a < b < c} .
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow} Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is convex on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} . Then for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \in [0, 1]} we have that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad f(tx + (1 - t)y) \leq tf(x) + (1 - t)f(y) \end{align}}
- Set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a = x} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b = tx + (1-t)y} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c = y} . Combining the first and third equations with the second equation gives us:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad b = ta + (1-t)c \end{align}}
- Solving for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} gives us:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad b = ta + c - tc \\ \quad b - c = t(a - c) \\ \end{align}}
- Therefore:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad t = \frac{b - c}{a - c} = \frac{c - b}{c - a} \end{align}}
- Similarly, we compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 - t} to be:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 1 - t = \frac{c - a}{c - a} - \frac{c - b}{c - a} = \frac{b - a}{c - a} \end{align}}
- From the convexity of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(tx + (1-t)y) \leq tf(x) + (1-t)f(y)} , or equivalently:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad f(b) \leq \frac{c - b}{c - a}f(a) + \frac{b - a}{c - a}f(c) \\ \end{align}}
- And hence:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (c - a)f(b) \leq (c - b)f(a) + (b - a)f(c) = (c - a + a - b)f(a) + (b - a)f(c) = (c - a)f(a) - (b - a)f(a) + (b - a)f(c) \end{align}}
- Therefore:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (c - a)[f(b) - f(a)] \leq (b - a)[f(c) - f(a)] \quad \Leftrightarrow \quad \frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(a)}{c - a} \end{align}}
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow} Obtained by working backwards from above. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}
We state yet another important definition for convex and concave functions.
- Theorem 2: Let >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : I \to \mathbb{R}} .
- a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is convex on >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} if and only if for all >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in I} with >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a <; b <; c} we have that >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(b)}{c - b}}} .
- b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is concave on >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} if and only if for all >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in I} with > we have that >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{\frac{f(b) - f(a)}{b - a} \geq \frac{f(c) - f(b)}{c - b}}} .
Theorem 2 gives us a nice characterization of convex functions. It tells us that a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : I \to \mathbb{R}} is convex if and only if whenever we take three points Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in I} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a <; b <; c} we have that the slope of the line connecting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a, f(a))} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b, f(b))} is less than or equal to the sope of the line connecting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (b, f(b))} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (c, f(c))} . In other words, the slope of the line segments connecting consecutive pairs of points on the graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is increasing.
We can combine theorems 1 and 2 to get a nice chain of inequalities. That is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : I \to \mathbb{R}} is convex if and only if for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a, b, c \in I} with >Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a < b < c} we have that:
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{f(b) - f(a)}{b - a} \leq \frac{f(c) - f(a)}{c - a} \leq \frac{f(c) - f(b)}{c - b} \end{align}}
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