Chapter 5 Theorems

Extreme Value Theorem (EVT)

• if f is continous on a closed interval [a,b] then f has an absolute maximum and an absolute minimum on [a,b]

• It does not say where, so end point values play no role in determining if it is true

Condition for a Local Maximum or Minimum

• if f has a local max or local min at c then f’(c) does not exist or f'(c) = 0 at that point.

• The converse is false: ex. x³ has f’(0)=0 but no extremum at 0 does make it a candidate though (critical number)

Fermat’s Theorem

• f is differentiable and has a local max or min at c, then f’(c)=0

Locating Absolute Extreme Values (Candidates Test)

• If f is continuous on a closed interval [a, b], then the absolute max and min are the largest and smallest of f at the critical numbers in (a, b) and at a and b

• Evaluate f at EVERY candidate and write the table not a sign chart

Critical Number

A number c in the domain of f where f’(c) = 0 or f’(c) does not exist.

• A local extremum can not occur at an end point because it needs an open interval around c

Rolle’s Theorem

• if f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there must be at least one c in (a,b) such that f’(c) = 0

• All three conditions are required

Mean Value Theorem

• if f is continuous on [a,b] and differentiable on (a,b), then there is at least on c in (a,b) with f’(c) = slope of the secant line between a and b

• Remember: c is the x value not the slope

Corollary: Zero Derivative

• if f is continuous on [a, b],differentiable on (a, b), and f'(x) =0 for all x in (a, b), then f is constant on (a, b)

• this one is really just common sense

Corollary: Equal Derivatives

• if f and g are differentiable on (a.b) and f’(x) = g’(x) for all x in (a,b), then f(x)=g(x) + C on (a,b)

• this one is really just common sense

Increasing and Decreasing Function Test

• if f is differentiable on (a,b), with f’(x)>0 (or f’(x) < 0) on (a,b), then f is increasing (or decreasing). If f is also continous on [a,b] then this extends to [a.b]

First Derivative Test

• if f is continuous on an interval I, c is a critical number, and f is differentiable on an open interval (a,b) around c (except possibly at c)

• then f’ changes + to - at c: local max. f' changes - to + at c: local min. Same sign on both sides: neither

Test for Concavity

• if f is continuous on [a,b], and f’ and f’’ exist on (a,b), then f’’ (x) > 0 on (a, b): concave up. f’’ (x) < 0 on (a, b): concave down

• from the sign of f’’

Condition for an Inflection Point

• f is differentiable on an open interval containing c, and (c, f(c)) is an inflection point

• f’’ (c) = 0 or f’’ (c) does not exist

• Basically the critical point of the f’ graph

Second Derivative Test

• if f’ and f’’ exist on an open interval (a,b) containing c, and f’(c) = 0

• f’’(c) < 0: local max

• f”” (c) > 0: local min

• If f’’ (c)= 0 or does not exist, the test gives no information. Use the First Derivative Test instead