(a) Explanation Given Information: The function is g ( x ) = x − 2 x 2 − 1 and the interval is 0 ≤ x < 1 . Calculation: Point of local maximum . The point at which the derivative of function changes positive value to negative value is called local maximum point. Point of local minimum . The point at which the derivative of function changes negative value to positive value is called local minimum point. Provide the function as follows. g ( x ) = x − 2 x 2 − 1 (1) Calculate the first derivative of g ( x ) . g ′ ( x ) = ( x 2 − 1 ) ( 1 ) − ( x − 2 ) ( 2 x ) ( x 2 − 1 ) 2 = x 2 − 1 − 2 x 2 + 4 x ( x 2 − 1 ) 2 = − x 2 + 4 x − 1 ( x 2 − 1 ) 2 (2) Calculate the critical point of function f . At the critical point, the first derivative of f is zero. − x 2 + 4 x − 1 ( x 2 − 1 ) 2 = 0 x 2 − 4 x + 1 = 0 (3) The root of Equation (3) is the critical point of function g . Calculate theroot of Equation (3). x = 2 − 3 in 0 ≤ x < 1 The behaviour of g according to sign pattern of g ′ as shown in Table 1. Sign of g ′ Behaviour of g + (Positive) Increasing ‑ (Negative) Decreasing The sign pattern of reveals the behavior of g between and at the critical points, as summarized in the Table 2 (b) To determine Graph the function in the given domain and calculate any absolute extreme values are there.

3rd Edition

ISBN: 9780134175706

Author: Unknown

Publisher: PEARSON

See similar textbooks

Videos

Question

Chapter 4.3, Problem 55E

(a)

To determine

Calculate the function’s local extreme values and the point of occurrence.

(b)

To determine

Graph the function in the given domain and calculate any absolute extreme values are there.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 4.3, Problem 54E

Chapter 4.3, Problem 56E

Students have asked these similar questions

Calculus III May I please have the example, definition semicolons, and all blanks completed and solved? Thank you so much,

A company estimates that the revenue (in dollars) from the sale of x doghouses is given by R(x) = 12,000 In (0.02x+1). Use the differential to approximate the change in revenue from the sale of one more doghouse if 80 doghouses have already been sold. The revenue will increase by $ if one more doghouse is made. (Round to the nearest cent as needed.)

The population of bacteria (in millions) in a certain culture x hours after an experimental 20x nutrient is introduced into the culture is P(x) = - 2 Use the differential to approximate the changes in population for the following changes in x. 8+x a. 1 to 1.5 b. 3 to 3.25 a. Use the differential to approximate the change in population for x=1 to 1.5. Between 1 and 1.5 hours, the population of bacteria changes by million. (Round to three decimal places as needed.)

Chapter 4 Solutions

University Calculus

Ch. 4.1 - In Exercises 16, determine from the graph whether...Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 3E Ch. 4.1 - In Exercises 1–6, determine from the graph whether...Ch. 4.1 - Prob. 5E Ch. 4.1 - Prob. 6E Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - In Exercises 7–10, find the absolute extreme...Ch. 4.1 - Prob. 9E Ch. 4.1 - Prob. 10E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.