# Notes Nov 30, 2020 Vocabulary and Theorems

?

## Vocabulary

• Mean Value Theorem
• Critical Number/Critical Point
• Extreme (Absolute/relative max/min)
• Extreme Value Theorem (MVT)
1 of 5

## Mean Value Theorem (MVT)

Conditions: If

(1) f(x) is continuous on [a,b]

(2) f(x) is differetiable on (a,b)

Then: there exists an x=c where

- f'(c) = f(b) - f(a)/ b-a

- Instantaneous rate of change = average rate of change

- Doesn't tell us what C is. It only tells us that there is at least one number C that will satisfy the conclusion of the theorem.

* if f'(x)=0 for all x in an interval (a,b), then on f(x) is constant on (a,b)

*f'(x) = g'(x) for all x in an interval (a,b) then is this interval we have f(x)=g(x)+c where C is some constant

2 of 5

## Critical Number (Critical Point)

"C" (f(c) is definded)

is an x-value on an interval where either

- f'(c) = 0 or

- f'(c) is undefined

we say that x=c is s critical point of the function f(x) if f(c) exists is either of the following are true.

*We require that f(c) exists in order to actually be a critical point.

3 of 5

## Extreme: Minimum or Maximum

relative (local) - x=c is a local min

if f(c) is lower than or equal to all points in its neighborhood

absolute (global)

x=c is an absolute max/min if f(c) is (larger/smaller) then /or equal to) all points in the domain

If x=c is a relative extreme, then, x=c is the critical number

False: if x=c is a critical number, then x=c is a relative extreme

*Extreme value: give the y-value or height

(1) we say that f(x) has an absolute (or global) maximum at x=c if f(x)-< f(c) for every x in the domain we are working on

(2) we say that f(x) has a relative (or local) maximum at x=c if f(x) -< f(c) for every x in some open interval around x=c

(3) We say that f(x) has an absolute (or global) minimum at x=c if f(x)->

4 of 5

## Extreme Value Theorem

If f(x) is continuous and defined on [a,b]

Then there will exist at least one absolute max and min

5 of 5