Using the derivative of f(x) given below, determine the intervals on which f(x) is increasing or decreasing.
-$$f ^ { \prime } ( x ) = x ^ { 1 / 3 } ( x - 7 )$$
A) Decreasing on $( 0,7 )$; increasing on $( - \infty , 0 ) \cup ( 7 , \infty )$
B) Increasing on $( 0 , \infty )$
C) Decreasing on $( 0,7 )$; increasing on $( 7 , \infty )$
D) Decreasing on $( - \infty , 0 ) \cup ( 7 , \infty )$; increasing on $( 0,7 )$

Question

Quizplus · Accepted Answer

To determine where $f(x)$ is increasing or decreasing, we look at the sign of $f'(x) = x^{1/3}(x - 7)$. This derivative is zero when $x = 0$ and $x = 7$, dividing the real line into three intervals: $(-\infty, 0)$, $(0, 7)$, and $(7, \infty)$. For $x < 0$, $x^{1/3}$ is negative and $x - 7$ is negative, making $f'(x)$ positive, so $f(x)$ is increasing on $(-\infty, 0)$. For $0 < x < 7$, $x^{1/3}$ is positive and $x - 7$ is negative, making $f'(x)$ negative, so $f(x)$ is decreasing on $(0, 7)$. For $x > 7$, both $x^{1/3}$ and $x - 7$ are positive, making $f'(x)$ positive, so $f(x)$ is increasing on $(7, \infty)$.

Using the Derivative of F(x) Given Below, Determine the Intervals f′(x)=x1/3(x−7)f ^ { \prime } ( x ) = x ^ { 1 / 3 } ( x - 7 )f′(x)=x1/3(x−7)

Using the Derivative of F(x) Given Below, Determine the Intervals $f ^ { \prime } ( x ) = x ^ { 1 / 3 } ( x - 7 )$