How do I find the absolute maximum and minimum of a function?

Question

Cora Alexander · Accepted Answer

You find where the derivative is zero

STEP ONE
Find two things:

 1. The endpoints
 2. The places where the slope is zero

The end points could be the maximum or minimum because we don't know where the function starts or finishes
If the slope is zero, you know that there is a relative maximum or minimum.  We need to check if it is the absolute maximum or minimum or not.
 

#|color (lime) "by setting the derivatives of the function to zero" #
The derivative is a function of the slope. Thus, if you can see the derivative to zero, you can solve for all the relative max or min points.
STEP TWO
Plug each possible max or min point into the original function (not the derivative because we do not care about the slope anymore), and see which one is the largest and which one is the smallest.
Some function have multiple absolute maximums and minimums, especially trig functions.

Cora Alexander · Answer

undefined To find the absolute maximum and minimum of a function $ f(x) $ on a closed interval $[a, b]$:

1. Find all critical points of $ f(x) $ within the interval $[a, b]$. Critical points are where the derivative $ f'(x) $ equals zero or is undefined.

2. Evaluate the function at each critical point and at the endpoints of the interval $[a, b]$.

3. The largest of these values is the absolute maximum, and the smallest is the absolute minimum.

4. If the function is continuous on the closed interval $[a, b]$ and differentiable on the open interval $ (a, b) $, then the absolute maximum and minimum may also occur at the endpoints of the interval.

5. If the function is not continuous or not differentiable on the interval, then you must also check any points of discontinuity within the interval.

6. Once you've found the critical points and evaluated the function at those points and at the endpoints of the interval, compare the values to determine the absolute maximum and minimum.