MODULE 5: OPTIMIZATION

Second Step to finding the absolute extrema of f on a closed interval

Evaluate f at the critical numbers and at the endpoints of the closed interval, a and b

An extrema occurs at every critical number.

False

If x = 2 is a critical number, then f'(2) = 0.

False, a critical number is where the derivative equals zero or is undefined

If f'(5) = 0, then there is a maximum or minimum at x = 5.

False, not all critical points are extrema

First Step to finding the absolute extrema of f on a closed interval

Find the critical numbers of f in (a, b)

Extreme Value Theorem

If f is continuous on [a,b] then f has an absolute maximum and an absolute minimum on [a,b]. The global extrema occur at critical points in the interval or at endpoints of the interval.

Third Step to finding the absolute extrema of f on a closed interval

The smallest value is the absolute minimum and the largest value is the absolute maximum

When looking for extrema, where do you find the candidates for the "candidates test"?

all critical points and endpoints of the closed interval

Step 1 of Optimization

convert the constraint into a one variable equation

If f '(c) = 0 or f '(c) is undefined, then c is called

critical number of f

Step 4 of Optimization

equal that derivative to zero to get the critical numbers

f(c) is the absolute (global) minimum if

f(c) ≤ f(x) for all x in the domain

f(c) is the absolute (global) maximum if

f(c) ≥ f(x) for all x in the domain

What is needed to apply the extreme value theorem?

function must both be continuous on the closed interval to guarantee both absolute extrema

When being asked to find the value of the derivative at a certain indicated extremum,

plug the critical number(s) into the derivative

When being asked to locate the absolute extrema of the function on the closed interval,

plug the critical number(s) into the original equation

Step 5 of Optimization

plug the critical numbers back into the constraint equation

Step 2 of Optimization

plug the resulting constraint equation into the maximize/minimize formula

Step 6 of Optimization

plug those ending values into the maximize/minimize equation to get the final answer

Step 3 of Optimization

take the derivative of the new maximize/minimize formula

Constraint

use what is given in the problem to create this

Maximize/Minimize

use what you are trying to solve for to create this equation