Find all relative extrema of the function $f ( x ) = - 2 x ^ { 2 } - 8 x - 12$ . Use the Second Derivative Test where applicable.
A)relative max: $f ( - 2 ) = - 4$
B)relative min: $f ( 0 ) = - 12$
C)no relative min
D)no relative max
E)both A and C
F) both B and D

Question

Quizplus · Accepted Answer

First, find the derivative of $f(x) = -2x^2 - 8x - 12$, which is $f'(x) = -4x - 8$. Setting $f'(x) = 0$ gives $x = -2$. The second derivative, $f''(x) = -4$, is negative, indicating a concave down graph at $x = -2$, thus $x = -2$ is a relative maximum. There is no relative minimum because the parabola opens downwards.

Find All Relative Extrema of the Function f(x)=−2x2−8x−12f ( x ) = - 2 x ^ { 2 } - 8 x - 12f(x)=−2x2−8x−12

Find All Relative Extrema of the Function $f ( x ) = - 2 x ^ { 2 } - 8 x - 12$