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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 4.30
Chapter 4, Problem 4.30

Estimate the open intervals on which the function y = Ζ’(𝓍) is


a. increasing.
b. decreasing.
c. Use the given graph of Ζ’' to indicate where any local extreme
values of the function occur, and whether each extreme
is a relative maximum or minimum.
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1
Identify the intervals where the derivative Ζ’'(𝓍) is positive. These intervals correspond to where the function y = Ζ’(𝓍) is increasing.
Identify the intervals where the derivative Ζ’'(𝓍) is negative. These intervals correspond to where the function y = Ζ’(𝓍) is decreasing.
Locate the points where the derivative Ζ’'(𝓍) changes sign from positive to negative. These points are potential locations for relative maxima of the function y = Ζ’(𝓍).
Locate the points where the derivative Ζ’'(𝓍) changes sign from negative to positive. These points are potential locations for relative minima of the function y = Ζ’(𝓍).
Use the graph of Ζ’'(𝓍) to confirm the exact locations of these sign changes and determine the nature of each local extremum (maximum or minimum) based on the sign change.

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Key Concepts

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

Derivative and its Sign

The derivative of a function, denoted as Ζ’'(𝓍), provides information about the rate of change of the function. If Ζ’'(𝓍) is positive over an interval, the function is increasing on that interval. Conversely, if Ζ’'(𝓍) is negative, the function is decreasing. Understanding the sign of the derivative is crucial for determining where the function is increasing or decreasing.
Recommended video:
05:44
Derivatives

Critical Points

Critical points occur where the derivative Ζ’'(𝓍) is zero or undefined. These points are potential locations for local extreme values, such as relative maxima or minima. Analyzing the behavior of the derivative around these points helps identify whether they correspond to peaks (maxima) or troughs (minima) in the function.
Recommended video:
04:50
Critical Points

First Derivative Test

The First Derivative Test is used to classify critical points as relative maxima or minima. By examining the sign change of Ζ’'(𝓍) around a critical point, one can determine the nature of the extremum. If Ζ’'(𝓍) changes from positive to negative, the point is a relative maximum; if it changes from negative to positive, it is a relative minimum.
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The First Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

35. Determine the dimensions of the rectangle of largest area that can be inscribed in the right triangle shown in the accompanying figure.

Textbook Question

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


f(x) = x(4 βˆ’ x)Β³

Textbook Question

Roots (Zeros)


Show that the functions in Exercises 19–26 have exactly one zero in the given interval.


r(ΞΈ) = 2ΞΈ βˆ’ cosΒ²ΞΈ + √2, (βˆ’βˆž, ∞)

Textbook Question

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) = {βˆ’x, 0 ≀ x < 1

x βˆ’ 1, 1 ≀ x ≀ 2

Textbook Question

93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \(\y\)'' as positive, negative, or zero.

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Textbook Question

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dΒ²y/dxΒ² = 2 βˆ’ 6x; yβ€²(0) = 4, y(0) = 1