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.

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.)
b. the function is never increasing.

Explanation:

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

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(f(x)=2x^{2}-16x + 4\), the derivative \(f^\prime(x)=(2x^{2})^\prime-(16x)^\prime+(4)^\prime\).
\(f^\prime(x)=4x-16\).

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 intervals of increase and decrease

Choose a test - point in the interval \((-\infty,4)\), say \(x = 0\). Then \(f^\prime(0)=4\times0 - 16=-16<0\). So \(f(x)\) is decreasing on \((-\infty,4)\).
Choose a test - point in the interval \((4,\infty)\), say \(x = 5\). Then \(f^\prime(5)=4\times5-16 = 4>0\). So \(f(x)\) is increasing on \((4,\infty)\).

Answer:

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