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