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
(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.

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 \(x\):
\(8x=16\), then \(x = 2\).

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

• For \(x<2\) (e.g., \(x = 1\)), \(f^{\prime}(1)=8\times1-16=-8<0\).
• For \(x>2\) (e.g., \(x = 3\)), \(f^{\prime}(3)=8\times3 - 16=8>0\).

Since \(f^{\prime}(x)<0\) when \(x<2\), the function is decreasing on the interval \((-\infty,2)\).

Answer:

A. The function is decreasing on \((-\infty,2)\)