Identify the open intervals where the function $f ( x ) = x \sqrt { 30 - x ^ { 2 } }$
is increasing or decreasing.
A) decreasing: $( - \infty , \sqrt { 15 } )$; increasing: $( \sqrt { 15 } , \infty )$
B) decreasing on $( - \infty , \infty )$
C) increasing: $( - \infty , \sqrt { 30 } )$; decreasing: $( \sqrt { 30 } , \infty )$
D) increasing: $( - \sqrt { 15 } , \sqrt { 15 } ) ;$ decreasing: $( - \sqrt { 30 } , - \sqrt { 15 } ) \cup ( \sqrt { 15 } , \sqrt { 30 } )$
E) increasing: $( - \sqrt { 30 } , \sqrt { 15 } ) \cup ( \sqrt { 15 } , \sqrt { 30 } ) ;$ decreasing: $( - \sqrt { 15 } , \sqrt { 15 } )$

Question

Identify the open intervals where the function $f ( x ) = x \sqrt { 30 - x ^ { 2 } }$
is increasing or decreasing. 
A) decreasing: $( - \infty , \sqrt { 15 } )$; increasing: $( \sqrt { 15 } , \infty )$
B) decreasing on $( - \infty , \infty )$
C) increasing: $( - \infty , \sqrt { 30 } )$; decreasing: $( \sqrt { 30 } , \infty )$
D) increasing: $( - \sqrt { 15 } , \sqrt { 15 } ) ;$ decreasing: $( - \sqrt { 30 } , - \sqrt { 15 } ) \cup ( \sqrt { 15 } , \sqrt { 30 } )$
E) increasing: $( - \sqrt { 30 } , \sqrt { 15 } ) \cup ( \sqrt { 15 } , \sqrt { 30 } ) ;$ decreasing: $( - \sqrt { 15 } , \sqrt { 15 } )$

Quizplus · Accepted Answer

To find the intervals where the function is increasing or decreasing, we need to compute the derivative of the function:

$$f'(x) = \sqrt{30-x^2} -\frac{x^2}{\sqrt{30-x^2}}$$

We need to study the sign of $f'(x)$ to identify the intervals where the function is increasing or decreasing. Let's factorize:

$$f'(x) = \frac{(30-x^2) - x^2}{\sqrt{30-x^2}} = \frac{30-2x^2}{\sqrt{30-x^2}}$$

The numerator is zero for $x=\pm\sqrt{15}$, and the denominator is positive for $x\in(-\infty,-\sqrt{30})\cup(\sqrt{30},\infty)$ and undefined for $x\in\{-\sqrt{30},\sqrt{30}\}$. We can use a sign chart to determine the sign of $f'(x)$ in each interval:

$$\begin{array}{|c||c|c|c|c|c|} \hline x & (-\infty,-\sqrt{30}) & -\sqrt{30} & (-\sqrt{30},-\sqrt{15}) & -\sqrt{15} & (-\sqrt{15},\sqrt{15}) \ \hline f'(x) & + & 	ext{undefined} & - & 0 & + \ \hline x & (\sqrt{15},\sqrt{30}) & \sqrt{30} & (\sqrt{30},\infty) & & \ \hline f'(x) & - & 	ext{undefined} & + & & \ \hline \end{array} $$

Therefore, the function is increasing in the intervals $(-\sqrt{15},\sqrt{15})$ and decreasing in the intervals $(-\infty,-\sqrt{30})\cup(-\sqrt{15},\sqrt{15})\cup(\sqrt{15},\sqrt{30})$. This matches choice D.

Identify the Open Intervals Where the Function f(x)=x30−x2f ( x ) = x \sqrt { 30 - x ^ { 2 } }f(x)=x30−x2​

Identify the Open Intervals Where the Function $f ( x ) = x \sqrt { 30 - x ^ { 2 } }$