Extreme Value Theorem

​Extreme Value of a Function, Mean Value Theorem, and Rolle Theorem

"​Reasoning is the Seasoning that Flavors your Life."

By. Mr. C.

Multiple Select

In order for the Extreme Value theorem to apply, which of these must be true. Select all that apply. #1

• It must be discontinuous

• It must be closed

• It must be open

• It must be continuous

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Multiple Select

Where can critical values occur? (check all that apply) #23

• Endpoints

• x-intercepts

• when f'(x)=0

• when f'(x) is undefined

Multiple Choice

When you check for sign changes at the critical points of a function, you are using the... #24

• IVT

• MVT

• 1st Derivative Test

• 2nd Derivative Test

Multiple Choice

If  $$f'\left(a\right)=0$$  and  $$f'\left(x\right)$$  changes from positive to negative at  $$x=a$$  , then  $$f\left(x\right)$$  has 

#25

• A relative maximum at x=a

• A relative minimum at x=a

• No relative extrema at x=a

• A vertical tangent line at x=a

Multiple Choice

If  $$f'\left(a\right)=0$$  and  $$f'\left(x\right)$$  changes from negative to positive at $$x=a$$  , then  $$f\left(x\right)$$  has

#26

• a relative maximum at x=a

• a relative minimum at x=a

• no relative extrema at x=a

• a vertical tangent line at x=a

Multiple Choice

For each problem, find the x-coordinates of all critical points. #29

• A

• B

• C

• D

Multiple Choice

Find the critical value(s) of

f(x) = x² + 2x + 1. #34

• x = -1

• x = 2

• x = -1, 0

• x = 0

Multiple Choice

Find the absolute extrema of

g(x) = 3x³ + 6x² on [-1, 1]. #35

• abs max value of $$\frac{32}{9}$$ when x = $$-\frac{4}{3}$$ abs min value of 0 when x = 0

• abs max value of 9 when x = 1, abs min value of 3 when x = -1

• abs max value of 9 when x = 1, abs min value of 0 when x = 0

• abs max value of 9 when x = 1, abs min of -1 when x = -1

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Multiple Choice

Which of these sums up the Mean Value Theorem (MVT)? #2

• $$f'\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

• $$f\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

• $$f\left(c\right)=\frac{f'\left(b\right)-f'\left(a\right)}{b-a}$$

• $$f'\left(c\right)=\frac{f'\left(b\right)-f'\left(a\right)}{b-a}$$

Multiple Choice

The geometrical interpretation of the Mean Value Theorem for the curve y =f(x) defined in [a,b] says ∃(there exists) #10

• a point c ∈ [a,b] where tangent when drawn is parallel to the chord joining end points of the curve y = f(x).

• a point c ∈ (a,b) where tangent when drawn is parallel to the X-axis.

• a point c ∈ (a,b) where tangent when drawn is parallel to the chord joining end points of the curve y = f(x).

• a point c ∈ [a,b] where tangent when drawn is parallel to the X-axis.

Multiple Choice

$$f\left(x\right)=\left(x-4\right)\left(x-6\right)\ \ \ \ \ \ \ \left[4,10\right].$$  Verify the Mean Value Theorem for the given f(x). #14

• c = 6

• c =7

• c = 5

• c = 3

Multiple Choice

The Mean Value Theorem applies to f(x) = 3x - x² on the interval [2, 5]. Find the value of x satisfying the Mean Value Theorem. #15

• 2

• -4

• 3.5

• -2

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Multiple Select

The geometrical interpretation of Rolle's theorem for y = f(x) in [a,b] is there exist #9

• a point c ∈ (a,b) where tangent when drawn is parallel to X-axis.

• a point c ∈ [a,b] where tangent when drawn is parallel to X-axis

• a point c ∈ (a,b) where tangent when drawn is parallel to the chord joining end points of the curve y = f(x).

• a point c ∈ [a,b] where tangent when drawn is parallel to the chord joining end points of the curve y = f(x).

Multiple Choice

Find the two x-intercepts of f(x) = x² – 12x + 20,  and then find c that verifies  Rolle’s Theorem within the interval. #46

• intercepts:

  x = 2, x = 10

  c = 6

• intercepts:

  x = -2, x = -10

  c = -6

• intercepts:

  x = 2, x = 10

  c = 4

• intercepts:

  x = -2, x = 10

  c = 6