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) = sin 2x, 0 ≤ x ≤ π

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 x − cos x,0 ≤ x ≤ 2π

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) = √3cos x + sin x, 0 ≤ x ≤ 2π

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

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) = csc²x − 2cot x, 0 < x < π

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) = sec²x − 2tan x, −π/2 < x < π/2

Identifying Extrema

In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.

Identifying Extrema

In Exercises 61 and 62, the graph of f' is given. Assume that f is continuous, and determine the x-values corresponding to local minima and local maxima.

Identifying Extrema

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

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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

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.

Identifying Extrema

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

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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

f(x) = x − 6√(x − 1)

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

f(x) = (x² − 3) / (x − 2), x ≠ 2

Identifying Extrema

In Exercises 19–40:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local extreme values, if any, saying where they occur.

f(x) = x¹ᐟ³(x + 8)

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

d. Determine all extrema of f.