Second Derivative Test

After establishing the critical points of a function, the second-derivative test uses the value of the second derivative at those points to determine whether such points are a local maximum or a local minimum. If the function f is twice-differentiable at a critical point x (i.e. a point where f '(x) = 0), then:

• If $f''(x)<0$, then $f$ has a local maximum at $x$.
• If $f''(x)>0$, then $f$ has a local minimum at $x$.
• If $f''(x)=0$, the test is inconclusive.

In the last case, Taylor's Theorem may sometimes be used to determine the behavior of f near x using higher derivatives.

Proof of the second-derivative test

Suppose we have $f''(x)>0$ (the proof for $f''(x)<0$ is analogous). By assumption, $f'(x)=0$. Then

$0<f''(x)=\lim _{h\to 0}{\frac {f'(x+h)-f'(x)}{h}}=\lim _{h\to 0}{\frac {f'(x+h)}{h}}.$

Thus, for h sufficiently small we get

${\frac {f'(x+h)}{h}}>0,$

which means that $f'(x+h)<0$ if $h<0$ (intuitively, f is decreasing as it approaches $x$ from the left), and that $f'(x+h)>0$ if $h>0$ (intuitively, f is increasing as we go right from x). Now, by the first-derivative test, $f$ has a local minimum at $x$.

Concavity test

A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point. It does not, however, provide information about inflection points. Specifically, a twice-differentiable function f is concave up if $f''(x)>0$ and concave down if $f''(x)<0$. Note that if $f(x)=x^{4}$, then $x=0$ has zero second derivative, yet is not an inflection point, so the second derivative alone does not give enough information to determine whether a given point is an inflection point.

Higher-order derivative test

The higher-order derivative test or general derivative test is able to determine whether a function's critical points are maxima, minima, or points of inflection for a wider variety of functions than the second-order derivative test. As shown below, the second-derivative test is mathematically identical to the special case of n = 1 in the higher-order derivative test.

Let f be a real-valued, sufficiently differentiable function on an interval $I\subset \mathbb {R}$, let $c\in I$, and let $n\geq 1$ be a natural number. Also let all the derivatives of f at c be zero up to and including the n-th derivative, but with the (n + 1)th derivative being non-zero:

$f'(c)=\cdots =f^{(n)}(c)=0\quad {\text{and}}\quad f^{(n+1)}(c)\neq 0.$

There are four possibilities, the first two cases where c is an extremum, the second two where c is a (local) saddle point:

• If n is odd and $f^{(n+1)}(c)<0$, then c is a local maximum.
• If n is odd and $f^{(n+1)}(c)>0$, then c is a local minimum.
• If n is even and $f^{(n+1)}(c)<0$, then c is a strictly decreasing point of inflection.
• If n is even and $f^{(n+1)}(c)>0$, then c is a strictly increasing point of inflection.

Since n must be either odd or even, this analytical test classifies any stationary point of f, so long as a nonzero derivative shows up eventually.

Example

Say, we want to perform the general derivative test on the function $f(x)=x^{6}+5$ at the point $x=0$. To do this, we calculate the derivatives of the function and then evaluate them at the point of interest until the result is nonzero.

$f'(x)=6x^{5}$, $f'(0)=0;$

$f''(x)=30x^{4}$, $f''(0)=0;$

$f^{(3)}(x)=120x^{3}$, $f^{(3)}(0)=0;$

$f^{(4)}(x)=360x^{2}$, $f^{(4)}(0)=0;$

$f^{(5)}(x)=720x$, $f^{(5)}(0)=0;$

$f^{(6)}(x)=720$, $f^{(6)}(0)=720.$

As shown above, at the point $x=0$, the function $x^{6}+5$ has all of its derivatives at 0 equal to 0, except for the 6th derivative, which is positive. Thus n = 5, and by the test, there is a local minimum at 0.

Resources

• Second Derivative Test, The Organic Chemistry Tutor

References

• Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (Third ed.). New York: McGraw-Hill. pp. 231–267. ISBN 0-07-010813-7.
• Marsden, Jerrold; Weinstein, Alan (1985). Calculus I (2nd ed.). New York: Springer. pp. 139–199. ISBN 0-387-90974-5.
• Shockley, James E. (1976). The Brief Calculus : with Applications in the Social Sciences (2nd ed.). New York: Holt, Rinehart & Winston. pp. 77–109. ISBN 0-03-089397-6.
• Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8.
• Willard, Stephen (1976). Calculus and its Applications. Boston: Prindle, Weber & Schmidt. pp. 103–145. ISBN 0-87150-203-8.