Chapter 4.3, Problem 54E

To determine

Whether the given statement is true or false along with justification.

Answer to Problem 54E

It has been determined and justified that the given statement is true.

Explanation of Solution

Given:

If $f^{\prime }\left(c\right)=0$ and $f^{″}\left(c\right)<0$ , then $f\left(c\right)$ is a local maximum.

Concept used:

According to the second derivative test:

If $f^{\prime }\left(c\right)=0$ and $f^{″}\left(c\right)<0$ , then $x=c$ is a point of local maxima and $f\left(c\right)$ is a local maximum.

If $f^{\prime }\left(c\right)=0$ and $f^{″}\left(c\right)>0$ , then $x=c$ is a point of local minima and $f\left(c\right)$ is a local minimum.

Calculation:

If $f^{\prime }\left(c\right)=0$ , then the slope of the tangent to the function at the point $x=c$ , is zero.

This implies that at $x=c$ , the tangent to the function is horizontal.

Now, if $f^{″}\left(c\right)<0$ , then the function is concave down at the point $x=c$ .

Combining, if the tangent to the function is horizontal and if the function is concave down at the point $x=c$ , then $x=c$ must be a point of local maxima.

Hence, if $f^{\prime }\left(c\right)=0$ and $f^{″}\left(c\right)<0$ , then $f\left(c\right)$ is a local maximum.

So, the given statement is true.

Conclusion:

It has been determined and justified that the given statement is true.