Question

use the second derivative test to classify the relative extrema of the following function if the test applies otherwise use the first derivative

Answer 1

Given the function

$\begin{gathered} f(x)=-6x^2-60x-96 \\ f^{\prime}(x)=-12x-60 \end{gathered}$

The second derivative of the function is obtained as

$\begin{gathered} f^{\doubleprime}(x)=-12 \\ \end{gathered}$

Since the f''(x) is negative, the function has a maxima.

At the critical point, f'(x)=0. Thus,

$\begin{gathered} f^{\prime}(x)=-12x-60 \\ -12x-60=0 \\ x=-5 \end{gathered}$

Substitute the value of x in the function, to obtain y.

$\begin{gathered} y=-6x^2-60x-96 \\ y=-6(-5)^2-60(-5)-96 \\ y=54 \end{gathered}$

Thus, we obtain a relative maxima at (-5,54)

There is no relative minimum.