Question

find the intervals on which ( f(x) ) is increasing, the intervals on which ( f(x) ) is decreasing, and the local extrema.

( f(x)=4 x^{2}-16 x + 4 )

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function is increasing on ( (2, infty) ).
(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)
b. the function is never increasing.

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. the function is decreasing on ( (-infty, 2) ).
(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)
b. the function is never decreasing.

select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
(type integers or simplified fractions.)

a. the function has a local minimum ( f(square)=square ), and no local maximum.
b. the function has a local maximum ( f(square)=square ) and a local minimum ( f(square)=square ).
c. the function has a local maximum ( f(square)=square ), and no local minimum.
d. the function has no local extrema.

Explanation:

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

The derivative of \(f(x)=4x^{2}-16x + 4\) is \(f^\prime(x)=8x-16\) using the power rule \((x^n)^\prime=nx^{n - 1}\).

Step2: Find the critical points

Set \(f^\prime(x)=0\), so \(8x-16 = 0\). Solving for \(x\):

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Step3: Determine the sign of \(f^\prime(x)\) in intervals

• For \(x<2\) (e.g., \(x = 1\)), \(f^\prime(1)=8\times1-16=-8<0\). So \(f(x)\) is decreasing on \((-\infty,2)\).
• For \(x>2\) (e.g., \(x = 3\)), \(f^\prime(3)=8\times3 - 16=8>0\). So \(f(x)\) is increasing on \((2,\infty)\).

Step4: Find the local extremum

Since the function changes from decreasing to increasing at \(x = 2\), we find \(f(2)\):

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Answer:

• For the increasing - interval: A. The function is increasing on \((2,\infty)\)
• For the decreasing - interval: A. The function is decreasing on \((-\infty,2)\)
• For the local extremum: A. The function has a local minimum \(f(2)=-12\), and no local maximum.