Hubble Space Telescope
The Hubble Space Telescope was deployed on April 24, 1990 by the space shuttle Discovery. A model for velocity of the shuttle during this mission, from liftoff at t = 0 until the solid rocket boosters were jettisoned at t = 126 sec, is given by: v(t) = 0.0013t3 – 0.09t2 + 23.6t – 3.083 (in ft/sec). Using this model, estimate the absolute maximum and minimum values of the acceleration of shuttle between liftoff and the jettisoning of the boosters.
Sol: We are asked for the extreme values not of the given velocity function, but rather of the acceleration function. So we need to differentiate the velocity function to find the acceleration: We now apply the closed interval method to the continuous function a on the interval 0 ≤ t ≤ 126 Its derivative is: a'(t) = 0.0078t – 0.18 The only critical no. occurs when a'(t) = 0 is: = 23.08 Evaluating a(t) at the critical number and the endpoints, we have a(0) = 23.6, a(23.08) = 21.52 and a(126) = 62.84 So the maximum acceleration is about 62.84 ft/s2 and the minimum acceleration is about 21.52 ft/s2.

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

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.

Example