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Maxima & Minima

## Absolute (Global) Extreme Values

Introduction: One of the most useful things we can learn from a function's derivative is whether the function assumes any maximum or minimum values on a given interval and where these values are located if it does. Once we know how to find a function's extreme values, we will be able to answer such questions as "What is the maximum acceleration of a space shuttle?" and "What is the radius of a contracted windpipe that expels air most rapidly during a cough?".

Absolute (Global) extreme values: A function f has an absolute or global maximum at 'c' if f(c) ≥ f(x) for all 'x' in domain of the function f. The number f(c) is called the absolute maximum value of f on its domain. Similarly, f has an absolute or global minimum at 'c' if f(c) ≤ f(x) for all 'x' in domain of the function f. The number f(c) is called the absolute minimum value of f on its domain.

Together, the absolute minimum and the absolute maximum are known as the absolute (global) extrema of the function. We often skip the term absolute or global and just say maximum and minimum values of the function.

Following figure shows the graph of a function f with absolute maximum at 'c' and absolute minimum at 'a'. The value of f at 'a', that is, f(a) is called the absolute minimum value and the value of f at 'c', that is, f(c) is called the absolute maximum value of the function f.

The extreme value theorem: If the function f is continuous on a closed interval [a, b], then f attains the maximum value f(c) and the minimum value f(d) at some numbers 'c' and 'd' in the interval [a, b].

Closed interval method
To find an absolute extrema of a continuous function f on a closed interval [a, b]:

• Find the values of f at the critical numbers [a number in the interior of the domain of a function f(x) at which = 0 or does not exist] of f in (a, b).
• Find the values of f at the endpoints of the interval.
• The largest of the values from above two steps is the absolute maximum value of f; the smallest of these values is the absolute minimum value of f.

Local (Relative) Extreme Values

Let 'c' be an interior point of the domain of the function f. Then f(c) is a local (relative) maximum at 'c' if and only if f(c) ≥ f(x) when 'x' is near 'c'. This means that f(c) ≥ f(x) for all 'x' in some open interval containing 'c'. Similarly, f(c) is a local (relative) minimum at 'c' if and only if f(c) ≤ f(x) when 'x' is near 'c'.

Together, the local (relative) minimum and the local (relative) maximum are known as the local (relative) extrema of the function.

Following figure shows the graph of a function f with local maximum at 'c' and local minimum at 'd'. The value of f at 'd', that is, f(d) is called the local minimum value and the value of f at 'c', that is, f(c) is called the local maximum value of the function f.

Fermat's theorem: If the function f has a local maximum or minimum at an interior point 'c' of its domain, and if exists at 'c', then (c) = 0.

From this theorem, we usually need to look at only a few points to find a function's extrema. These consist of the interior domain points where = 0 or does not exist and the domain endpoints.

In terms of critical numbers, Fermat's theorem can be rephrased as: if the function f has a local minimum or maximum at 'c', then 'c' is a critical number of f.

• Ex: Find relative extrema of the function f(x) = x3 – 3x + 6 in [–2, 3].

Sol: The graph of the function f(x) = x3 – 3x + 6 in [–2, 3] is shown in the figure below. From this graph, we can conclude that f(–1) = 8 is a local maximum, whereas f(1) = 4 is a local minimum.

The First Derivative Test and Concavity
First derivative test for relative extrema
This test applies to a continuous function f(x).
• At a critical point 'c', if changes from positive to negative, i.e., > 0 for x < c and < 0 for x > c, then the function f(x) has a local maximum value at 'c'.
• At a critical point 'c', if changes from negative to positive, i.e., < 0 for x < c and > 0 for x > c, then the function f(x) has a local minimum value at 'c'.
• At a critical point 'c', if does not change its sign, i.e., has the same sign on both sides of 'c', then the function f(x) has no local maximum or local minimum at 'c'.
• At a left end-point 'a', if > 0 [or < 0] for x > a, then f(x) has a local minimum [or maximum] value at 'a'.
• At a right end-point 'b', if > 0 [or < 0] for x < b, then f(x) has a local maximum [or minimum] value at 'b'.
First derivative test for absolute extrema
Suppose that 'c' is a critical number of a continuous function f(x) defined on an interval.
• If > 0 for all x < c and < 0 for all x > c, then f(c) is the absolute maximum value of f(x).
• If < 0 for all x < c and > 0 for all x > c, then f(c) is the absolute minimum value of f(x).

Concavity: If the graph of a function f lies above all of its tangents on an interval, then it is called the concave upward on the interval. If the graph of a function f lies below all of its tangents on an interval, then it is called the concave downward on the interval.

Following figure shows the graph of a function f that is concave downward on the interval (a, b) and the concave upward on the interval (b, c).

Test for concavity
Let f be a twice differentiable function.
• If (x) > 0 for all x in interval I, then the graph of f is concave upward on I.
• If (x) < 0 for all x in interval I, then the graph of f is concave downward on I.