# 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 ${\displaystyle f''(x)<0}$, then ${\displaystyle f}$ has a local maximum at ${\displaystyle x}$.
• If ${\displaystyle f''(x)>0}$, then ${\displaystyle f}$ has a local minimum at ${\displaystyle x}$.
• If ${\displaystyle f''(x)=0}$, 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 ${\displaystyle f''(x)>0}$ (the proof for ${\displaystyle f''(x)<0}$ is analogous). By assumption, ${\displaystyle f'(x)=0}$. Then

${\displaystyle 0

Thus, for h sufficiently small we get

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

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

## 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 ${\displaystyle f''(x)>0}$ and concave down if ${\displaystyle f''(x)<0}$. Note that if ${\displaystyle f(x)=x^{4}}$, then ${\displaystyle x=0}$ 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 ${\displaystyle I\subset \mathbb {R} }$, let ${\displaystyle c\in I}$, and let ${\displaystyle n\geq 1}$ 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:

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

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 ${\displaystyle f^{(n+1)}(c)<0}$, then c is a local maximum.
• If n is odd and ${\displaystyle f^{(n+1)}(c)>0}$, then c is a local minimum.
• If n is even and ${\displaystyle f^{(n+1)}(c)<0}$, then c is a strictly decreasing point of inflection.
• If n is even and ${\displaystyle f^{(n+1)}(c)>0}$, 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 ${\displaystyle f(x)=x^{6}+5}$ at the point ${\displaystyle x=0}$. To do this, we calculate the derivatives of the function and then evaluate them at the point of interest until the result is nonzero.

${\displaystyle f'(x)=6x^{5}}$, ${\displaystyle f'(0)=0;}$
${\displaystyle f''(x)=30x^{4}}$, ${\displaystyle f''(0)=0;}$
${\displaystyle f^{(3)}(x)=120x^{3}}$, ${\displaystyle f^{(3)}(0)=0;}$
${\displaystyle f^{(4)}(x)=360x^{2}}$, ${\displaystyle f^{(4)}(0)=0;}$
${\displaystyle f^{(5)}(x)=720x}$, ${\displaystyle f^{(5)}(0)=0;}$
${\displaystyle f^{(6)}(x)=720}$, ${\displaystyle f^{(6)}(0)=720.}$

As shown above, at the point ${\displaystyle x=0}$, the function ${\displaystyle x^{6}+5}$ 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.

## References

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• 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.
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