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

Using the Derivative to Test for Extremes

The first derivative of a function tells about the general shape of the function, and we can use that shape information to determine if an extreme point is a maximum or minimum or neither.

First Derivative Test for Local Extremes

Let $f$ be a continuous function with $f^{\prime}(a)=0$ or $f^{\prime}(a)$ is undefined.

(i) If $\mathrm{f}^{\prime}$ (left of $a$ ) $> 0$ and $\mathrm{f}'$ (right of $a$ ) $< 0$ , then $(\mathrm{a}, \mathrm{f}(\mathrm{a}))$ is a local maximum (Fig. 19a)

(ii) If $\mathrm{f}'$ (left of $a$ ) $< 0$ and $\mathrm{f}'$ (right of $a$ ) $> 0$ , then $(\mathrm{a}, \mathrm{f}(\mathrm{a}))$ is a local minimum (Fig. 19b)

(iii) If $\mathrm{f}^{\prime}$ (left of $a$ ) $> 0$ and $\mathrm{f}'$ (right of $a$ ) $> 0$ , then $(\mathrm{a}, \mathrm{f}(\mathrm{a}))$ is not a local extreme (Fig. 19c)

(iv) If $\mathrm{f}'$ (left of $a$ ) $< 0$ and $\mathrm{f}'$ (right of $a$ ) $< 0$ , then $(\mathrm{a}, \mathrm{f}(\mathrm{a}))$ is not a local extreme (Fig. 19d)

Fig. 19

Practice 9: Find all extremes of $f(x)=3 x^{2}-12 x+7$ and use the First Derivative Test to determine if they are maximums, minimums or neither.

A variant of the First Derivative Test can also be used to determine whether an endpoint gives a maximum or minimum for a function.