The Second Derivative and the Shape of a Function f(x)

The concavity of a function can also help us determine whether a critical point is a maximum or minimum or neither. For example, if a point is at the bottom of a concave up function (Fig. 7), then the point is a minimum.

Fig. 7

Proof: (a) Assume that $\mathrm{f}^{\prime}(\mathrm{c})=0$. If $\mathrm{f}^{\prime \prime}(\mathrm{c}) < 0$ then $\mathrm{f}$ is concave down at $\mathrm{x}=\mathrm{c}$ so the graph of $\mathrm{f}$ will be below the tangent line $L(x)$ for values of $x$ near $c$. The tangent line, however, is given by $L(x)=f(c)+f^{\prime}(c)(x-c)=f(c)+0(x-c)=f(c)$, so if $x$ is close to $c$ then $f(x) < L(x)=f(c)$ and $f$ has a local maximum at $\mathrm{x}=\mathrm{c}$. The proof of (b) for a local minimum of $\mathrm{f}$ is similar.

(c) If $\mathrm{f}^{\prime}(\mathrm{c})=0$ and $\mathrm{f}^{\prime \prime}(\mathrm{c})=0$, then we cannot immediately conclude anything about local maximums or minimums of $f$ at $x=c$. The functions $f(x)=x^{4}, g(x)=-x^{4}$, and $h(x)=x^{3}$ all have their first and second derivatives equal to zero at $x=0$, but $f$ has a local minimum at $0, g$ has a local maximum at $0$, and $\mathrm{h}$ has neither a local maximum nor a local minimum at $\mathrm{x}=0$.

The Second Derivative Test for Extremes is very useful when $\mathrm{f}^{\prime \prime}$ is easy to calculate and evaluate. Sometimes, however, the First Derivative Test or simply a graph of the function is an easier way to determine if we have a local maximum or a local minimum – it depends on the function and on which tools you have available to help you.

Practice 2: $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^{3}-15 \mathrm{x}^{2}+24 \mathrm{x}-7$ has critical numbers $\mathrm{x}=1$ and $4$. Use the Second Derivative Test for Extremes to determine whether $\mathrm{f}(1)$ and $\mathrm{f}(4)$ are maximums or minimums or neither.