Objectives: In this tutorial, we investigate what derivatives tell us about local maxima and local minima. We apply these results to finding maxima and minima of functions having only one critical points and functions which are continuous on a closed interval. After working through these materials, the student should be able

• to find the local maxima and local minima of functions symbolically;
• to find the maximum and minimum of a function on a closed interval.
Modules:

• Review the definitions of maximum, minimum, local maximum and local minimum.

 Theorem. Suppose that a is a local maximum or a local minimum of the function f. Then either f '(a) = 0 or f '(a) does not exist.

• Discussion [Using Flash]

• Examples.

 Definition. Suppose that a is a point in the domain of the function f. a is a critical point of f if either f '(a) = 0 or f '(a) does not exist.

• The following theorem will be useful in later work.

 Theorem. Suppose that the function f is differentiable on an interval I and suppose that the point a in I is the only critical point of f. if a is a local maximum of f then a is a maximum of f, and if a is a local minimum of f then a is a minimum of f.

• Example.

• The maximum and minimum of the function f(x) = x on the closed interval [1, 5] are the endpoints of the interval and are not critical points of f.

• A very important property of continuous functions is the following theorem.

 Theorem. Suppose that the function f is continuous on the closed interval [a, b]. Then there exists c in [a, b] such that c is a maximum of f, and there exists d in [a, b] such that d is a minimum of f. We note that c and d are either critical points of f or endpoints of [a, b].

• Examples.

• Find the maximum and minimum of a function of the form f(x) = a x2 + b x + c on a closed interval.
Discussion.
• Find the maximum and minimum of a function of the form f(x) = a x3 + b x2 + c x + d on a closed interval.
Discussion.

• Drill problems on finding maxima and minima of functions on closed intervals.