# Maximum/Minimum values of a graph on a restricted interval

## Homework Statement

f(x) = x^3 + 12x^2 - 27x + 11
Absolute Maximum
Absolute Minimum
on the interval [-10,0]
(there are 3 different interval sets, but if I can do this one, I think I can figure out the rest.)

## Homework Equations

Derivative, set equal to 0, then solve for the problem, but what I'm confused about is how the solving process differs as the interval changes.

## The Attempt at a Solution

I have the derivative set as
3x^2 + 24x - 27

but what I'm unsure about is how finding the absolute maximum on the interval [-10,0] differs in process from finding the interval on, say, [-7, 2]

I think what I'm really trying to ask is how the restrictions on the intervals are reflected in the mathematical process to solve for an absolute maximum and minimum.

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LCKurtz
Homework Helper
Gold Member
The extremes of a continuous function on a closed interval must occur for values of x on the interval where one of the following is true:

1. f'(x) = 0
2. f'(x) fails to exist.
3. x is an end point of the closed interval.

You have to check all three cases, noting that values of x that aren't on the interval are irrelevant. Just list the possibilities and pick out the max and min.

edit: ah, someone just posted earlier delete some part of my post