5. Graphical Applications of Derivatives

Theory and Examples

Sketch the graph of a differentiable function y = f(x) that has a local maxima at (1, 1) and (3, 3)

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = 2x³ - 15x² + 24x on [0,5]

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x³ -4a²x

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = -2x⁴ + x² + 10

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

g(x) = x² − 4x + 4, 1 ≤ x < ∞

Determine where the local and absolute maxima and minima occur on the given graph of $f\(\left\)(x\(\right\))$.

37. What value of a makes f(x) = x^2 +(a/x) have

a. a local minimum at x = 2?

b. a point of inflection at x = 1?