## Absolute Extreme Values

One of the most useful things we can learn from a function's derivative is whether the function assumes any maximum or minimum value on a given interval and if it does, where these values are located. Once we know how to find a function's extreme values, we will be able to answer questions such as:

"What is the maximum acceleration of a space shuttle?"

"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', i.e., f(a) is called the absolute minimum value and the value of **f** at 'c', i.e., 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 |
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To find an absolute extrema of a continuous function f on a closed interval [a, b]:i. 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).ii. Find the values of f at the endpoints of the interval.iii. 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. |

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', i.e., f(d) is called the local minimum value and the value of **f** at 'c', i.e., 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) = x^{3} – 3x + 6 in [–2, 3].

**Sol:** The graph of the function f(x) = x^{3} – 3x + 6 in [–2, 3] is shown in the figure below.

From the graph, we can conclude that f(–1) = 8 is a local maximum, whereas f(1) = 4 is a local minimum.

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 |
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Let f be a twice differentiable function. |

i. If (x) > 0 for all x in interval I, then the graph of f is concave upward on I. |

ii. If (x) < 0 for all x in interval I, then the graph of f is concave downward on I. |