Lesson12Extrema

Question 1

What are the extrema on an interval?

Answer 1

They are the highest y-value and the lowest y-value on the x-interval.

Question 2

When do the extrema occur?

Answer 2

(1) When the derivative of the function is 0
(2) When the derivative is undefined.
(3) At one or both of the endpoints

Question 3

What are the extrema of

f(x) = x² - 2x on the closed interval [0,4]?

Answer 3

f’(x) = 2x - 2

at x=1, it has a horizontal tangent

It could be 1 or one of the endpoints:

^{(You evaluate at the original function:)}

at 0, y=0; at 4, y=8

So, 1 must be a low point

Question 4

What is the Extreme Value Theorem?

Answer 4

If a function is continuous on a closed interval, it has a minimum and a maximum value on that interval.

Question 5

What are local extrema?

How do you express it?

Answer 5

These are other points on the graph that have derivatives of 0, which are not the overall extrema.

“The relative minimum value is -4, and it occurs at x=2.

Question 6

What are Critical Numbers of a function?

Answer 6

• ^{Critical numbers are where “something” changes in a function.}*
  (1) derivative is 0 (a relative (local) minimum or maximum)
  (2) derivative is undefined (ex. a sharp corner)
  (3) on closed intervals, they also include the endpoints

Question 7

Are all critical numbers minimums or maximums?

Answer 7

No. Example: f(x) = x³

f’(x) = 3x²

Question 8

What are the absolute max and min values of

f(x) = 2x - 3x^⅔

on the interval [-1,3]?

Answer 8

f’(x) = 2 - 2x^-⅓

Critical numbers = -1, 3, 1(^{derivative= 0)} and 0^{(derivative doesn’t exist)}

Min: x = -1, y = -5

Max: x = 0, y = 0

Question 9

What are the guidelines for finding extrema on a closed interval?

Answer 9

(1) Find the critical #s on the open interval (must be in the domain)
(2) Evaluate (find the y) at the critical #s and the endpoints
(3) Select the winners

Question 10

Give an example of a potential critical number that is not in the domain.

Answer 10

f(x) = 1/x

f’(x) = -x^-2

You may think that 0 is a critical # because there is no derivative, but 0 was never in the domain because of the original function.

Question 11

Find the critical #s of

f(x) = x⁴ - 4x²

Answer 11

f’(x) = 4x³ - 8x

= 4x(x² - 2)

= 0 when x = 0 and when x² = 2

Critical #s = 0 ±√2

Question 12

Find the critical #s of:

f(x) = sin²x + cosx

at (0,2π)

Answer 12

f’(x) = 2sinxcosx - sinx

= sinx(2cosx - 1)

sinx=0 at π

cosx=½ at ^π/₃ and ^5π/₃

Question 13

Find the absolute extrema of y= -x² + 3x - 5

at [-2,1]

Answer 13

y’(x) = -2x + 3

x=0 at 3/2 which is not in the interval

endpoints =(-2,-15) abs. mim

and (1,-3) abs max