At what value of x does the function $f ( x ) = 3 x - \sqrt [ 3 ] { x }$ change from increasing to decreasing?
A) $\frac { 1 } { 8 }$
B) $\frac { 1 } { 2 }$
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 27 }$
E) $\frac { 1 } { 9 }$
F) $\frac { \sqrt { 3 } } { 4 }$
G) $\frac { \sqrt { 3 } } { 3 }$
H) $\frac { 1 } { 32 }$

Question

At what value of x does the function $f ( x ) = 3 x - \sqrt [ 3 ] { x }$ change from increasing to decreasing?
A) $\frac { 1 } { 8 }$ 
B) $\frac { 1 } { 2 }$ 
C) $\frac { 1 } { 3 }$
D) $\frac { 1 } { 27 }$ 
E) $\frac { 1 } { 9 }$ 
F) $\frac { \sqrt { 3 } } { 4 }$  
G) $\frac { \sqrt { 3 } } { 3 }$
H) $\frac { 1 } { 32 }$

Quizplus · Accepted Answer

The function changes from increasing to decreasing at a critical point where the derivative is zero. The derivative of $f(x) = 3x - \sqrt[3]{x}$ is $f'(x) = 3 - \frac{1}{3}x^{-2/3}$. Setting $f'(x) = 0$ gives $3 = \frac{1}{3}x^{-2/3}$, solving this equation gives $x = \frac{1}{27}$.

At What Value of X Does the Function f(x)=3x−x3f ( x ) = 3 x - \sqrt [ 3 ] { x }f(x)=3x−3x​

At What Value of X Does the Function $f ( x ) = 3 x - \sqrt [ 3 ] { x }$