Page 84 - How To Solve Word Problems In Calculus

If f is continuous on a closed bounded interval I = [a, b],
then f has both an absolute maximum and an absolute
minimum in I.

Of course, f may have an absolute maximum and/or minimum
if the function fails to be continuous or if I is not closed, but
there is no guarantee of their existence unless both hypotheses
of the extreme value theorem are satisfied.

A critical number for f is a number x for which either

f (x) = 0or f (x) does not exist.

In a word problem arising from a physical or geometrical sit-
uation it is very rare that f (x) will fail to exist. Therefore, in

this book we will consider only critical numbers for which
f (x) = 0.

It is easily shown that if a function is continuous on a
closed bounded interval, then its absolute extrema (which ex-
ist by the Extreme Value Theorem) will occur either at a critical
number or at an endpoint of the interval. The following pro-
cedure, which we will call the “closed interval method,” can
be used to determine their values.

Step1
Find all critical numbers of f (x) that lie within the given
interval [a, b]. Critical numbers outside the interval may be
ignored.

Step2
Compute the values of f (x) at each critical number and
at the endpoints a and b.

Step3
The largest value of f (x) obtained in step 2 is the ab-
solute maximum and the smallest value is the absolute
minimum.

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