Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Finding Global Extrema

Calculus

5. Graphical Applications of Derivatives

Finding Global Extrema

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4:37 minutes

Problem 4.1.39

Textbook Question

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

Verified step by step guidance

First, understand that finding absolute extrema on a closed interval involves evaluating the function at critical points and endpoints of the interval. The function given is g(θ) = θ³ᐟ⁵, and the interval is [-32, 1].

To find critical points, calculate the derivative of the function g(θ). The derivative, g'(θ), can be found using the power rule for derivatives. For g(θ) = θ³ᐟ⁵, the derivative is g'(θ) = (3/5)θ⁻²ᐟ⁵.

Set the derivative g'(θ) equal to zero to find critical points. Solve the equation (3/5)θ⁻²ᐟ⁵ = 0. Since the derivative is never zero for any real θ, there are no critical points from setting the derivative to zero.

Next, evaluate the function g(θ) at the endpoints of the interval. Calculate g(-32) and g(1) to determine the values of the function at these points.

Compare the values of g(θ) at the endpoints to determine the absolute maximum and minimum values of the function on the interval [-32, 1]. The largest value is the absolute maximum, and the smallest value is the absolute minimum.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Absolute Extrema

Absolute extrema refer to the highest and lowest values a function can achieve on a given interval. To find these values, evaluate the function at critical points and endpoints of the interval. The absolute maximum is the largest value, and the absolute minimum is the smallest value within the specified range.

Critical Points

Critical points are values of the variable where the derivative of the function is zero or undefined. These points are potential candidates for local maxima or minima. In the context of finding absolute extrema, critical points within the interval are evaluated alongside the endpoints to determine the function's extreme values.

Evaluating Function at Endpoints

Evaluating the function at the endpoints of the interval is crucial when determining absolute extrema. The endpoints are the boundary values of the interval, and the function's value at these points must be compared with values at critical points to identify the absolute maximum and minimum values on the interval.

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Textbook Question

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) = x+ cos⁻¹x on [-1,1]

Textbook Question

{Use of Tech} Absolute maxima and minima

a. Find the critical points of f on the given interval.

b. Determine the absolute extreme values of f on the given interval.

c. Use a graphing utility to confirm your conclusions.

f(x) = 2ᶻ sin x on [-2,6]

Textbook Question

{Use of Tech} Critical points and extreme values

a. Find the critical points of the following functions on the given interval. Use a root finder, if necessary.

b. Use a graphing utility to determine whether the critical points correspond to local maxima, local minima, or neither.

c. Find the absolute maximum and minimum values on the given interval, if they exist

h(x) (5-x)/(x² + 2x - 3) on [-10,10]

Textbook Question

Theory and Examples

Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

Textbook Question

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

Textbook Question

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

Textbook Question

Absolute Extrema on Finite Closed Intervals

h(x) = ³√x, −1 ≤ x ≤ 8

Textbook Question

Absolute Extrema on Finite Closed Intervals

g(x) = √(4 − x²), −2 ≤ x ≤ 1

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