Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.3.33
Chapter 4, Problem 4.3.33

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.


g(x) = x√8 − x²

Verified step by step guidance
1
To find the intervals where the function is increasing or decreasing, first find the derivative of the function g(x) = x√(8 - x²). Use the product rule and chain rule to differentiate.
Set the derivative equal to zero to find the critical points. These points will help determine where the function changes from increasing to decreasing or vice versa.
Analyze the sign of the derivative on the intervals determined by the critical points. If the derivative is positive on an interval, the function is increasing there; if negative, the function is decreasing.
To identify local extrema, evaluate the function at the critical points. Use the first derivative test to determine if these points are local maxima or minima.
Summarize the intervals of increase and decrease, and list any local extrema along with their locations based on the analysis of the derivative and critical points.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5m
Was this helpful?

Key Concepts

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

Derivative and Critical Points

The derivative of a function provides information about its rate of change. To find where a function is increasing or decreasing, calculate its derivative and identify critical points where the derivative is zero or undefined. These points are potential locations for local extrema and help determine intervals of increase or decrease.
Recommended video:
04:50
Critical Points

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive over that interval, and decreasing if the derivative is negative. By analyzing the sign of the derivative across different intervals, you can determine where the function rises or falls, which is crucial for identifying behavior patterns in the function.
Recommended video:
07:32
Determining Where a Function is Increasing & Decreasing

Local Extrema

Local extrema refer to the local maximum or minimum values of a function within a specific interval. These occur at critical points where the derivative changes sign. To identify local extrema, evaluate the function at critical points and use the first or second derivative test to confirm whether these points are maxima or minima.
Recommended video:
05:58
Finding Extrema Graphically
Related Practice
Textbook Question

Each of Exercises 67–88 gives the first derivative of a continuous function y=f(x). Find y'' and then use Steps 2–4 of the graphing procedure described in this section to sketch the general shape of the graph of f.

69. y' = x(x - 3)²

107
views
Textbook Question

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.


y = (𝓍 + 1) / (𝓍² + 2𝓍 + 2)

212
views
Textbook Question

6. You are planning to close off a corner of the first quadrant with a line segment 20 units long running from (a, 0) to (0,b). Show that the area of the triangle enclosed by the segment is largest when a = b.

213
views
Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dy/dx = 2x − 7, y(2) = 0

17
views
Textbook Question

Finding Functions from Derivatives


Suppose that f(0) = 5 and that f'(x) = 2 for all x. Must f(x) = 2x + 5 for all x? Give reasons for your answer.

205
views
Textbook Question

Finding Indefinite Integrals


In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.


∫(√x + ³√x) dx

16
views