Extreme Value Theorem

It must be discontinuous

It must be closed

It must be open

It must be continuous

Endpoints

x-intercepts

when f'(x)=0

when f'(x) is undefined

IVT

MVT

1st Derivative Test

2nd Derivative Test

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

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

A

B

C

D

x = -1

x = 2

x = -1, 0

x = 0

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

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

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

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

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

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.

c = 6

c =7

c = 5

c = 3

2

-4

3.5

-2

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).

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