Local Maxima and Minima, Slides of Calculus

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Local Maxima and Minima

October 22, 2013

What derivatives tell us about a function and its graph

Plot the graph of the function f (x) = x^3 − 9 x^2 − 48 x + 52.

What derivatives tell us about a function and its graph

Plot the graph of the function f (x) = x^3 − 9 x^2 − 48 x + 52. The graph of f ′(x) tells us some information of the graph of f (x). When is f ′(x) > 0? when is f ′(x) < 0?

What derivatives tell us about a function and its graph

Plot the graph of the function f (x) = x^3 − 9 x^2 − 48 x + 52. The graph of f ′(x) tells us some information of the graph of f (x). When is f ′(x) > 0? when is f ′(x) > 0?

Local maxima and minima

Definition Suppose p is a point in the domain of f (x): f has a local minimum at p if f (p) is less than or equal to the values of f for points near p. f has a local maximum at p if f (p) is greater than or equal to the values of f for points near p.

How do we detect a local maximum or minimum

Definition (Critical point) For any function f , a point p in the domain of f , where f ′(p) = 0 or f ′(x) is undefined is called a critical point of the function. In addition, the point (p, f (p)) on the graph of f is also called a critical point (of the graph). A critical value of f is the value, f (p), of the function at a critical point, p.

At a critical point where f ′(p) = 0, the tangent line to the graph at p is horizontal.

Example

Example

How do we detect a local maximum or minimum

The critical points divide the domain of f into intervals on which the sign of the derivative remains the same.

How do we detect a local maximum or minimum

The critical points divide the domain of f into intervals on which the sign of the derivative remains the same. Therefore, if f is defined on the interval between two successive critical points, its graph cannot change direction on that interval, it is either going up or it is going down.

First Derivative Test for Local Maxima and Minima

Suppose p is a critical point of a continuous function f. Then, as we go from left to right: If f changes from decreasing to increasing at p, then f has a local minimum at p.

p

f decreasingf’<0 f increasingf’>

p

f increasing f decreasingf’< f’>

First Derivative Test for Local Maxima and Minima

Suppose p is a critical point of a continuous function f. Then, as we go from left to right: If f changes from decreasing to increasing at p, then f has a local minimum at p. If f changes from increasing to decreasing at p, then f has a local maximum at p.

p

f decreasingf’<0 f increasingf’>

p

f increasing f decreasingf’< f’>

Second Derivative Test for Local Maxima and Minima

Suppose p is a critical point of a continuous function f , and f ′(p) = 0. If f is concave up at p, then f has a local minimum at p.

equivalent to If f ′′(p) > 0, then f has a local minimum at p.

p

f decreasingf’<0 f increasingf’>

p

f increasing f decreasingf’< f’>

Second Derivative Test for Local Maxima and Minima

Suppose p is a critical point of a continuous function f , and f ′(p) = 0. If f is concave up at p, then f has a local minimum at p. If f is concave down at p, then f has a local maximum at p. equivalent to If f ′′(p) > 0, then f has a local minimum at p. If f ′′(p) < 0, then f has a local maximum at p.

p

f decreasingf’<0 f increasingf’>

p

f increasing f decreasingf’< f’>