The second derivative test reveals that $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 8$ has
A) a relative maximum and two relative minima.
B) a relative maximum only.
C) a relative minimum only.
D) neither a relative maximum nor a relative minimum.

Question

Quizplus · Accepted Answer

The second derivative test involves finding the first and second derivatives of the function, then analyzing the critical points (where the first derivative is zero) to determine the nature of these points using the second derivative. For $f(x) = x^4 - 2x^2 + 8$, the first derivative $f'(x) = 4x^3 - 4x$, and setting $f'(x) = 0$ gives $x = 0, \pm 1$. The second derivative $f''(x) = 12x^2 - 4$. Evaluating $f''(x)$ at the critical points, we find $f''(0) = -4$, indicating a relative maximum at $x = 0$, and $f''(\pm 1) = 8$, indicating relative minima at $x = \pm 1$.

The Second Derivative Test Reveals That f(x)=x4−2x2+8f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 8f(x)=x4−2x2+8

The Second Derivative Test Reveals That $f ( x ) = x ^ { 4 } - 2 x ^ { 2 } + 8$