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Graphs of Functions and Derivatives

Extreme Values of Functions

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.

Real life application of Rolle's theorem Let's apply Rolle's theorem to the position function s = f(t), where f(t) is continuous and differentiable along its path. The function f(t) relates to a bullet fired from a war tank vertically upward with an initial velocity 40 m/sec.

After it is released from the tank at t = 0, it would reach the same place at t = 8 seconds. Therefore, f(0) = f(8). The bullet reaches its maximum height at t = 4 where the vertical velocity, that is, f '(t) becomes zero which is in the interval (0, 8).
Rolle's Theorem and Mean Value Theorem

Rolle's theorem: It was first published in 1691 by the French mathematician Michel Rolle. This theorem states that: if a function f is continuous on the closed interval [a, b], differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one real number 'c' in the open interval (a, b) such that (c) = 0.

From Rolle's theorem, we can see that if a function f is continuous on [a, b] and differentiable on (a, b), and if f(a) = f(b), there must be at least one x-value between a and b at which the graph of f has a horizontal tangent, as shown in above figure.

Mean value theorem: It was first stated by another French mathematician, Joseph-Louis Lagrange. This theorem states that: if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one number 'c' in (a, b) such that .

Geometrical interpretation of the mean value theorem: Geometrically, the theorem says that somewhere between points A and B on a differentiable curve, there is at least one tangent line parallel to secant line AB.

Physical interpretation of the mean value theorem: If we think of as the average rate of change of the function f over the interval [a, b] and (c) as an instantaneous rate of change, then the mean value theorem says that there must be a point in the open interval (a, b) at which the instantaneous rate of change is equal to the average rate of change over the interval [a, b].

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