What are the absolute extrema of f(x)=6x3−9x2−36x+3∈[−4,8]?

1 Answer
Roy D.
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 ddx[f(x)]=0 , use power rule:

ddx[6x3−9x2−36x+3]=18x2−18x−36

solve for the roots:
18x2−18x−36=0
x2−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)

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