5. Graphical Applications of Derivatives

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π

Estimate the open intervals on which the function y = ƒ(𝓍) is

a. increasing.

b. decreasing.

c. Use the given graph of ƒ' to indicate where any local extreme

values of the function occur, and whether each extreme

is a relative maximum or minimum.

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Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = sin 2x, 0 ≤ x ≤ π

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

ƒ(x) = x² √(x + 5)

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The function f(x) = √x has a local maximum on the interval [0,∞).

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

ƒ(x) = x³ / 3 - 9x

101. In Exercises 101 and 102, the graph of f' is given. Determine x-values corresponding to local minima, local maxima, and inflection points for the graph of f.

Explain how to apply the First Derivative Test.

Identifying Extrema

In Exercises 41–52:

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

k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0

Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.

a. Evaluate g(2), h(2), g'(2), and h'(2).

Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.

b. Does either g or h have a local extreme value at x = 2? Explain.

A tangent line approximation of a function value is an underestimate when the function is:

110. Suppose the derivative of the function y = f(x) is

y'=(x-1)^22(x-2)(x-4).

At what points, if any, does the graph of f have a local minimum, local maximum, or

point of inflection?

Identifying Extrema

In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).

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b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.

Finding Extrema from Graphs

In Exercises 11–14, match the table with a graph.