Find all relative extrema of the function $f ( x ) = x ^ { 2 / 3 } - 6$ . Use the Second Derivative Test where applicable.
A) relative minimum: $( 0 , - 6 )$
B) relative minimum: $( 0 , - 5 )$
C) relative maximum: $( 0 , - 6 )$
D) relative minimum: $( 0,2 )$
E) relative maximum: $( 0,5 )$

Question

Find all relative extrema of the function $f ( x ) = x ^ { 2 / 3 } - 6$ . Use the Second Derivative Test where applicable. 
A) relative minimum: $( 0 , - 6 )$
B) relative minimum: $( 0 , - 5 )$
C) relative maximum: $( 0 , - 6 )$
D) relative minimum: $( 0,2 )$
E) relative maximum: $( 0,5 )$

Quizplus · Accepted Answer

To find relative extrema, we first find the derivative of $f(x) = x^{2/3} - 6$, which is $f'(x) = \frac{2}{3}x^{-1/3}$. Setting $f'(x) = 0$ does not yield any solutions since $x^{-1/3}$ does not equal 0 for any real x. However, $f'(x)$ is undefined at $x = 0$, indicating a potential extremum. The second derivative, $f''(x) = -\frac{2}{9}x^{-4/3}$, evaluates to negative for $x > 0$ and positive for $x < 0$, indicating a concavity change at $x = 0$. However, the second derivative test is inconclusive here due to the nature of the function at $x = 0$. Observing the function, it decreases for $x < 0$ and increases for $x > 0$, suggesting a relative minimum at $x = 0$. Substituting $x = 0$ into the original function gives $f(0) = 0^{2/3} - 6 = -6$, thus the relative minimum is at $(0, -6)$.

Find All Relative Extrema of the Function f(x)=x2/3−6f ( x ) = x ^ { 2 / 3 } - 6f(x)=x2/3−6

Find All Relative Extrema of the Function $f ( x ) = x ^ { 2 / 3 } - 6$