Question

step 1
the absolute maximum and minimum values of f occur either at a critical point inside the interval or at an endpoint of the interval. recall that a critical point is a point where f(x)=0 or is undefined. we begin by finding the derivative of f.
f(x)=18x^{2}-36x - 54
step 2
we now solve f(x)=0 for x, which gives the following critical numbers. (enter your answers as a comma - separated list.)
x =

Explanation:

Step1: Solve the quadratic equation for critical - points

Set \(f^{\prime}(x)=18x^{2}-36x - 54 = 0\). First, divide the entire equation by 18: \(x^{2}-2x - 3=0\).

Step2: Factor the quadratic equation

Factor \(x^{2}-2x - 3\) as \((x - 3)(x+1)=0\).

Step3: Find the values of \(x\)

Set each factor equal to zero: \(x - 3=0\) gives \(x = 3\), and \(x + 1=0\) gives \(x=-1\).

Answer:

\(-1,3\)