Question

summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of ( y = f(x) ).
( f(x)=2 x^{4}-12 x^{2} )
(type your answer in interval notation. use a comma to separate answers as needed.)
what is the ( y )-intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the ( y )-intercept is ( y = 0 ).
b. there is no ( y )-intercept.
what is/are the ( x )-intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the ( x )-intercept(s) is/are ( x = -sqrt{6}, 0, sqrt{6} ).
(type an exact answer, using radicals as needed. use a comma to separate answers as needed.)
b. there is no ( x )-intercept.
what is/are the local maximum/a? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the local maximum/a is/are at ( x = ).
(simplify your answer. type an exact answer using radicals as needed. use integers or fractions for any numbers in the expression. use a comma
b. there is no local maximum.

Explanation:

Step1: Find the derivative of \(f(x)\)

The function is \(f(x)=2x^{4}-12x^{2}\). Using the power rule \((x^{n})^\prime = nx^{n - 1}\), the derivative \(f^\prime(x)=8x^{3}-24x=8x(x^{2}-3)=8x(x-\sqrt{3})(x + \sqrt{3})\)

Step2: Find critical points

Set \(f^\prime(x)=0\). So \(8x(x-\sqrt{3})(x+\sqrt{3})=0\). The critical points are \(x = 0,x=\sqrt{3},x=-\sqrt{3}\)

Step3: Use the second - derivative test

The second - derivative \(f^{\prime\prime}(x)=24x^{2}-24\)

• For \(x = 0\): \(f^{\prime\prime}(0)=24\times0^{2}-24=-24<0\). So \(x = 0\) is a local maximum.
• For \(x=\sqrt{3}\): \(f^{\prime\prime}(\sqrt{3})=24\times(\sqrt{3})^{2}-24=72 - 24 = 48>0\). So \(x=\sqrt{3}\) is a local minimum.
• For \(x=-\sqrt{3}\): \(f^{\prime\prime}(-\sqrt{3})=24\times(-\sqrt{3})^{2}-24=72 - 24 = 48>0\). So \(x=-\sqrt{3}\) is a local minimum.

Answer:

The local maximum is at \(x = 0\)