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 ) has an inflection point at ( x = -sqrt{5},sqrt{5} )
(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)
b. the function ( f ) has no inflection point.
choose the correct graph of ( y = f(x) ) below.

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 \(y = ax^{n}\), \(y^\prime=anx^{n - 1}\), 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 \(-12x^{2}+60 = 0\)
\(12x^{2}=60\), \(x^{2}=5\), \(x=\pm\sqrt{5}\)

Answer:

A. The function \(f\) has an inflection point at \(x =-\sqrt{5},\sqrt{5}\)