Find all relative extrema of the function $f ( x ) = x ^ { 8 / 9 } - 3$ . Use the Second Derivative Test where applicable.
A) relative max: $( 1 , - 2 )$; no relative min
B) no relative max or min
C) relative min: $( 0 , - 3 )$; no relative $\max$
D) relative max: $( 1 , - 2 )$; relative min: $( 0 , - 3 )$
E) relative max: $( 0 , - 3 )$; no relative min

Question

Find all relative extrema of the function $f ( x ) = x ^ { 8 / 9 } - 3$ . Use the Second Derivative Test where applicable. 
A) relative max: $( 1 , - 2 )$; no relative min
B) no relative max or min
C) relative min: $( 0 , - 3 )$; no relative $\max$
D) relative max: $( 1 , - 2 )$; relative min: $( 0 , - 3 )$
E) relative max: $( 0 , - 3 )$; no relative min

Quizplus · Accepted Answer

The first derivative $f'(x) = \frac{8}{9}x^{-1/9}$ is undefined at $x = 0$, indicating a critical point. The second derivative $f''(x) = -\frac{8}{81}x^{-10/9}$ is negative for $x > 0$ and positive for $x < 0$, indicating a relative minimum at $(0, -3)$.

Find All Relative Extrema of the Function f(x)=x8/9−3f ( x ) = x ^ { 8 / 9 } - 3f(x)=x8/9−3

Find All Relative Extrema of the Function $f ( x ) = x ^ { 8 / 9 } - 3$