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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.1.52d
Capitolo 4, Problema 4.1.52d

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


d. Determine all extrema of f.

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To find the extrema of the function \( f(x) = |x^3 - 9x| \), we first need to consider the critical points of the function. Critical points occur where the derivative is zero or undefined.
First, find the derivative of the inside function \( g(x) = x^3 - 9x \). The derivative is \( g'(x) = 3x^2 - 9 \).
Set \( g'(x) = 0 \) to find the critical points of \( g(x) \). Solve \( 3x^2 - 9 = 0 \) to find the values of \( x \) where the derivative is zero.
The absolute value function \( f(x) = |g(x)| \) can change its behavior at points where \( g(x) = 0 \). Solve \( x^3 - 9x = 0 \) to find these points.
Evaluate \( f(x) \) at the critical points and endpoints of the domain to determine the extrema. Consider the behavior of \( f(x) \) around these points to classify them as local minima, maxima, or neither.

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Absolute Value Function

The absolute value function, denoted as |x|, represents the distance of a number x from zero on the number line, always yielding a non-negative result. In the context of f(x) = |x³ − 9x|, it affects how we find extrema by considering both the positive and negative scenarios of the expression inside the absolute value.
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Average Value of a Function

Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential candidates for local extrema. For f(x) = |x³ − 9x|, we first need to find the derivative, set it to zero, and solve for x to identify critical points, considering the behavior of the absolute value function.
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Critical Points

Extrema

Extrema refer to the maximum and minimum values of a function. To determine extrema, evaluate the function at critical points and endpoints of the domain. For f(x) = |x³ − 9x|, analyze the critical points and the behavior of the function as x approaches positive and negative infinity to identify all local and global extrema.
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Finding Extrema Graphically
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Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[f(x) + g(x)] dx

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In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.

y=1-(x+1)^3

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The Mean Value Theorem                                                                                                                                                                  

                                                                                                                                                                                        

 a. Show that the equation 𝓍⁴ + 2𝓍² ― 2 = 0 has exactly one solution on [0,1] .

         

[Technology Exercises] b.Find the solution to as many decimal places as you can.  

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Domanda del libro di testo

Theory and Examples


In Exercises 51 and 52, give reasons for your answers.


Let f(x) = |x³ − 9x|.


b. Does f'(-3) exist?

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Domanda del libro di testo

Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[−f(x)] dx

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Domanda del libro di testo

106. Motion Along a Line The graphs in Exercises 105 and 106 show the position s=f(t) of an object moving up and down on a coordinate line. At approximately what times is the (d) When is the acceleration positive? Negative?

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