5. Graphical Applications of Derivatives

Absolute Extrema on Finite Closed Intervals

In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.

g(θ) = θ³ᐟ⁵, −32 ≤ θ ≤ 1

Absolute Extrema on Finite Closed Intervals

In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.

f(x) = x⁴ᐟ³, −1 ≤ x ≤ 8

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

f(x) = (2/3)x − 5, −2 ≤ x ≤ 3

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

h(x) = ³√x, −1 ≤ x ≤ 8

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

g(x) = √(4 − x²), −2 ≤ x ≤ 1

Absolute Extrema on Finite Closed Intervals

In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.

f(t) = 2 − |t|, −1 ≤ t ≤ 3

Finding Extrema from Graphs

In Exercises 1–6, determine from the graph whether the function has any absolute extreme values on [a, b]. Then explain how your answer is consistent with Theorem 1.

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

Finding Extrema from Graphs

In Exercises 7–10, find the absolute extreme values and where they occur.

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

f(x) = |x|, −1 < x < 2

Finding Extrema from Graphs

In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.

g(x) = {−x, 0 ≤ x < 1

x − 1, 1 ≤ x ≤ 2

Theory and Examples

[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.

f(x) = |x − 2| + |x + 3|, −5 ≤ x ≤ 5

Theory and Examples

[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.

h(x) = |x + 2| − |x − 3|, −∞ < x < ∞