Both a function and its derivative are given. Use them to find all critical numbers. $f ( x ) = x - 9 x ^ { 2 / 3 } + 7 \quad f ^ { \prime } ( x ) = \frac { x ^ { 1 / 3 } - 6 } { x ^ { 1 / 3 } }$
A) $x = 0$
B) $x = 216$
C) $x = 0 , x = - 101$
D) $x = 0 , x = 216$
E) $x = - 101 , x = 216$

Question

Quizplus · Accepted Answer

Critical numbers occur where the derivative is zero or undefined. The derivative $f'(x) = \frac{x^{1/3} - 6}{x^{1/3}}$ is undefined at $x = 0$ and equals zero when $x^{1/3} - 6 = 0$, which solves to $x = 216$.

Both a Function and Its Derivative Are Given f(x)=x−9x2/3+7f′(x)=x1/3−6x1/3f ( x ) = x - 9 x ^ { 2 / 3 } + 7 \quad f ^ { \prime } ( x ) = \frac { x ^ { 1 / 3 } - 6 } { x ^ { 1 / 3 } }f(x)=x−9x2/3+7f′(x)=x1/3x1/3−6​

Both a Function and Its Derivative Are Given $f ( x ) = x - 9 x ^ { 2 / 3 } + 7 \quad f ^ { \prime } ( x ) = \frac { x ^ { 1 / 3 } - 6 } { x ^ { 1 / 3 } }$