Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).
( f(x) = (x^{2}+6)(36 - x^{2}) )
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function ( f ) is concave upward on the subinterval(s) ( (-sqrt{5},sqrt{5}) ).
(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed)
b. the function ( f ) is never concave upward.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function ( f ) is concave downward on the subinterval(s) ( (-infty,-sqrt{5}),(sqrt{5},infty) ).
(type your answer in interval notation. type an exact answer, using radicals as needed. use a comma to separate answers as needed)
b. the function ( f ) is never concave downward.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the function ( f ) has an inflection point at ( x = ).
(type an exact answer, using radicals as needed. use a comma to separate answers as needed)
b. the function ( f ) has no inflection point.

Explanation:

Step1: Expand the function

$$\begin{align*} f(x)&=(x^{2}+6)(36 - x^{2})\\ &=36x^{2}-x^{4}+216 - 6x^{2}\\ &=-x^{4}+30x^{2}+216 \end{align*}$$

Step2: Find the first - derivative

Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(f^\prime(x)=-4x^{3}+60x\)

Step3: Find the second - derivative

Differentiate \(f^\prime(x)\) again. \(f^{\prime\prime}(x)=-12x^{2}+60\)

Step4: Find the inflection points

Set \(f^{\prime\prime}(x) = 0\), then \(-12x^{2}+60 = 0\).

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Step5: Determine concavity intervals

• For \(f^{\prime\prime}(x)>0\):

\(-12x^{2}+60>0\), \(x^{2}<5\), \(-\sqrt{5}

• For \(f^{\prime\prime}(x)<0\):

\(x^{2}>5\), \(x<-\sqrt{5}\) or \(x>\sqrt{5}\). The function is concave downward on \((-\infty,-\sqrt{5})\cup(\sqrt{5},\infty)\)

Answer:

For the concavity upward: A. The function \(f\) is concave upward on the sub - interval(s) \((-\sqrt{5},\sqrt{5})\)
For the concavity downward: A. The function \(f\) is concave downward on the sub - interval(s) \((-\infty,-\sqrt{5}),(\sqrt{5},\infty)\)
For the inflection points: A. The function \(f\) has an inflection point at \(x =-\sqrt{5},\sqrt{5}\)