Question

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

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

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

( \bigcirc ) a. the function is increasing on
(type your answer in interval notation. type integers or simplified fractions. use a comma to separate answers as needed.)

( \bigcirc ) b. the function is never increasing.

Explanation:

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

Given \(f(x)=2x^{2}-16x + 4\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have \(f^\prime(x)=\frac{d}{dx}(2x^{2})-\frac{d}{dx}(16x)+\frac{d}{dx}(4)\).
\(f^\prime(x)=4x-16\).

Step2: Find the critical points

Set \(f^\prime(x) = 0\), so \(4x-16=0\).
Add \(16\) to both sides: \(4x=16\).
Divide both sides by \(4\): \(x = 4\).

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

• For the interval \((-\infty,4)\), let's take a test - point \(x = 3\). Then \(f^\prime(3)=4\times3-16=12 - 16=-4<0\).
• For the interval \((4,\infty)\), let's take a test - point \(x = 5\). Then \(f^\prime(5)=4\times5-16=20 - 16 = 4>0\).

Answer:

A. The function is increasing on \((4,\infty)\)