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} )
on what interval(s) is ( f ) concave upward or concave downward?
(type your answer in interval notation. use a comma to separate answers as needed. use integers or fractions for any numbers in the expression.)
a. ( f ) is concave upward on ( (-infty,-1),(1, infty) ) and ( f ) is concave downward on ( (-1,1) )
b. ( f ) is never concave upward; ( f ) is concave downward on
c. ( f ) is never concave downward; ( f ) is concave upward on
sketch a graph of ( y = f(x) ). choose the correct graph below.

Explanation:

Step1: Find the first derivative

Using the power rule \((x^n)^\prime=nx^{n - 1}\), for \(y = f(x)=2x^{4}-12x^{2}\), the first derivative \(y^\prime=f^\prime(x)=8x^{3}-24x\).

Step2: Find the second derivative

Differentiate \(y^\prime = 8x^{3}-24x\) again. Using the power rule, \(y^{\prime\prime}=f^{\prime\prime}(x)=24x^{2}-24 = 24(x^{2}-1)=24(x + 1)(x - 1)\).

Step3: Determine concavity

Set \(y^{\prime\prime}=0\), then \(24(x + 1)(x - 1)=0\), which gives \(x=-1\) and \(x = 1\).

• For \(x\in(-\infty,-1)\), let \(x=-2\). Then \(y^{\prime\prime}(-2)=24((-2)^{2}-1)=24\times3>0\), so \(f(x)\) is concave upward on \((-\infty,-1)\).
• For \(x\in(-1,1)\), let \(x = 0\). Then \(y^{\prime\prime}(0)=24(0^{2}-1)=-24<0\), so \(f(x)\) is concave downward on \((-1,1)\).
• For \(x\in(1,\infty)\), let \(x = 2\). Then \(y^{\prime\prime}(2)=24(2^{2}-1)=24\times3>0\), so \(f(x)\) is concave upward on \((1,\infty)\).

Answer:

A. \(f\) is concave upward on \((-\infty,-1),(1,\infty)\) and \(f\) is concave downward on \((-1,1)\)