# Derivatives and Graphs

## Application of Derivatives to Graphs

Derivatives also help us to graph functions, by locating minimum, maximum, and inflection points. They can also determine the interval on which a function is convex or concave.

Convex - The graph is below the tangent lines. Example ${\displaystyle -x^{2}}$, think of a frown.

Concave - The graph is above the tangent lines. Example ${\displaystyle x^{2}}$, think of a smile.

Inflection Point - The point when a function changes from convex to concave or vice versa. Example ${\displaystyle x^{3}}$ at x = 0.

Maximum Point - The highest point of a function on a convex interval.

Minimum Point - The lowest point of a function on a concave interval.

Stationary Point - A point where f'(c) = 0

### Rules of Stationary Points

• If ${\displaystyle f'\left(c\right)=0}$ and ${\displaystyle f''\left(c\right)<0}$, then c is a local maximum point of f(x). The graph of f(x) will be concave on the interval.
• If ${\displaystyle f'\left(c\right)=0}$ and ${\displaystyle f''\left(c\right)>0}$, then c is a local minimum point of f(x). The graph of f(x) will be convex on the interval.
• If ${\displaystyle f'\left(c\right)=0}$ and ${\displaystyle f''\left(c\right)=0}$ and ${\displaystyle f'''\left(c\right)\neq 0}$, then c is a local inflection point of f(x).

### Locating And Evaluating Stationary Points

The Function

The original function is used to determine when the function crosses the x and y axis's.

The First Derivative Test

The first derivative is used to find all the minimum, maximum or inflection points. At these points f'(x) = 0. Also if we make a sign chart we can see at which intervals the function is increasing and decreasing, but the second derivative is a better test of this.

The Second Derivative Test

The second derivative is used to find the points when a function is concave or when it is convex at these points f''(x) = 0. Then you need to make a sign chart. The interval(s) that the sign of the second derivative is positive the function is concave and if the sign is negative the function is convex. When f'(a) is zero and on that interval f''(x) is negative then the point a is a maximum point, before this point the function will be increasing afterwards the function will be decreasing. When f'(a) is zero and on that interval f''(x) is positive then the point a is a minimum point, before this the function will be decreasing afterwards the function will be increasing.

The Third Derivative Test

The third derivative test determines if a point is an inflection point. If f''(c) = 0 and ${\displaystyle f'''(c)\neq 0}$, then f(c) is an inflection point.

#### An Example

Sketch the graph of ${\displaystyle f(x)=x^{3}-4x^{2}}$

1) The function will tell us that:

${\displaystyle f(x)=x^{2}(x-4)}$
${\displaystyle 0=x^{2}(x-4)}$ x = 0 and x=4 are the x-intercepts.
${\displaystyle y=0^{2}(0-4)}$ y = 0 is the y-intercept.

2) Then we use the first derivative test.

${\displaystyle f'(x)=3x^{2}-8x}$
${\displaystyle 0=x(3x-8)}$ x = 0 and x=2.66 are the minimum or maximum or inflection points.

3) Then we use the second derivative test.

f''(x) = 6x - 8
0 = 6x -8 so x = ${\displaystyle {\frac {4}{3}}}$.
We now make a number line __f'''(0) = -8______ 1.333____f'''(2) = 4.
Before 1.333 the function will be convex after 1.333 the function will be concave.

4) Then we use the third derivative test.

f'''(x) = 6
${\displaystyle f'''({\frac {4}{3}})=6}$ The point is an inflection point.

5) Now we gather all the information into a table. Also we need to find the corresponding y values to all the x values that we obtained from steps 1 to 3 using f(x).

Interval or Point f(x) behaviour
x < 0 Increasing
(0,0) Maximum Point, x-intercept, and y-intercept
0 < x < 1.33 Decreasing
(1.33,-4.74) Inflection Point
1.33 < x < 2.66 Decreasing
(2.66,-9.48) Minimum Point
2.66 < x < ${\displaystyle \infty }$ Increasing
(4,0) x-intercept

6)Now we can draw our graph: