What are the absolute extrema of f(x)= 6x^3 − 9x^2 − 36x + 3 in [-4,8]?

1 Answer
Jun 9, 2018

(-4,-381) and (8,2211)

Explanation:

In order to find the extrema, you need to take the derivative of the function and find the roots of the derivative.

i.e. solve for d/dx [f(x)] = 0 , use power rule:

d/dx [6x^3 - 9x^2-36x+3 ] = 18x^2-18x-36

solve for the roots:
18x^2-18x-36 = 0
x^2-x-2 = 0 , factor the quadratic:
(x-1)(x+2) = 0
x = 1, x = -2

f(-1) = -6-9+36+3 = 24
f(2) = 48-36-72+3 = -57

Check the bounds:
f(-4) = -381
f(8) = 2211

Thus the absolute extrema are (-4,-381) and (8,2211)