Question

Find the absolute maximum and absolute minimum values of f on the given interval.

f(x) = 4x³ - 18x² - 48x + 7, [-2, 5]

Answer 1

The first thing we need to do is evaluate the function at its endpoints.

`f(-2)=-1\\\\f(5)=-183`

Then, we need to find the critical points to find if the stationary points give the absolute min/max values.

`f'(x)=12x^2 - 36x-48=0\\\\x^2 - 3x-4=0\\\\(x-4)(x+1)=0\\ \\ x=-1, 4`

`f(-1)=33\\\\f(4)=-217`

So, the absolute maximum is 33 and the absolute minimum is -217