Calculus I: Extreme Values, Maxima & Minima, Mean Value Theorem, Summaries of Economics

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MTH4100 Calculus I

Bill Jackson School of Mathematical Sciences QMUL

Week 8, Semester 1, 2013

Extreme values of functions

Definition Let f be a function with domain D and c ∈ D. Then f has an absolute (or global) maximum value at c if f (x) ≤ f (c) for all x ∈ D. Similarly f has an absolute (or global) minimum value at c if f (x) ≥ f (c) for all x ∈ D. These values are collectively referred to as absolute extrema, or global extrema.

Local Maxima and Minima

These values are also called local extrema.

Local Maxima and Minima - Example

Note: Absolute extrema are automatically local extrema, but the converse need not be true.

First Derivative Theorem for Local Extrema

Theorem Suppose that f has a local maximum or minimum value at an interior point c of its domain, and that f is differentiable at c. Then f ′(c) = 0.

Note that the converse is false!

First Derivative Theorem for Local Extrema

Theorem Suppose that f has a local maximum or minimum value at an interior point c of its domain, and that f is differentiable at c. Then f ′(c) = 0.

Note that the converse is false! This theorem tells us that the extreme values of a function f can only occur at the following kinds of points: interior points of the domain where f ′^ = 0; interior points of the domain where f ′^ does not exist; endpoints of the domain.

Finding local extrema

The Extreme Value Theorem tells us that a continuous function f on a bounded closed interval has absolute maximum and minimum values. The First Derivative Theorem for Local Extrema gives us a method to determine these values:

Finding local extrema

The Extreme Value Theorem tells us that a continuous function f on a bounded closed interval has absolute maximum and minimum values. The First Derivative Theorem for Local Extrema gives us a method to determine these values: Step 1 Determine the citical points of f. Step 2 Evaluate f at each critical point AND at the end points of the interval. Step 3 Take the largest and smallest values appearing in Step 2.

Rolle’s theorem

This result tells us that a function which is continuous on a bounded closed interval and takes the same value at both endpoints of the interval must have at least one critical point in the interval.

Rolle’s theorem

This result tells us that a function which is continuous on a bounded closed interval and takes the same value at both endpoints of the interval must have at least one critical point in the interval. Theorem Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f (a) = f (b) then there exists a c ∈ (a, b) with f ′(c) = 0.

Rolle’s theorem - Note

It is essential that both the hypotheses in Rolle’s theorem are fulfilled i.e. f is continuous on [a, b] and differentiable on (a, b):

Rolle’s theorem - Note

It is essential that both the hypotheses in Rolle’s theorem are fulfilled i.e. f is continuous on [a, b] and differentiable on (a, b):

In each case there is no point c ∈ (a, b) with f ′(c) = 0.

Rolle’s theorem - Example

Apply Rolle’s theorem to f (x) = x 3 3 −^3 x^ on [−^3 ,^ 3].

The Mean Value Theorem

This extends Rolle’s theorem to the case when f (a) 6 = f (b).

Theorem (Mean Value Theorem) Let f (x) be continuous on [a, b] and differentiable on (a, b). Then there exists a c ∈ (a, b) with

f ′(c) =

f (b) − f (a) b − a