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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.3.31
Capitolo 4, Problema 4.3.31

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.


f(x) = x − 6√(x − 1)

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1
To determine where the function \( f(x) = x - 6\sqrt{x - 1} \) is increasing or decreasing, first find its derivative \( f'(x) \). Use the power rule and chain rule to differentiate: \( f'(x) = 1 - \frac{6}{2\sqrt{x - 1}} \). Simplify this to \( f'(x) = 1 - \frac{3}{\sqrt{x - 1}} \).
Identify the domain of \( f(x) \). Since \( \sqrt{x - 1} \) is defined for \( x \geq 1 \), the domain of \( f(x) \) is \( x > 1 \).
Find the critical points by setting \( f'(x) = 0 \). Solve \( 1 - \frac{3}{\sqrt{x - 1}} = 0 \) to find the critical points. This simplifies to \( \sqrt{x - 1} = 3 \), leading to \( x - 1 = 9 \), so \( x = 10 \).
Determine the intervals of increase and decrease by testing points in the intervals \( (1, 10) \) and \( (10, \infty) \). Choose a test point in each interval, such as \( x = 2 \) and \( x = 11 \), and evaluate \( f'(x) \) at these points to determine the sign of the derivative.
Identify the local extrema by analyzing the sign changes of \( f'(x) \). If \( f'(x) \) changes from positive to negative at \( x = 10 \), then \( f(x) \) has a local maximum at \( x = 10 \). If it changes from negative to positive, it would be a local minimum. Evaluate \( f(x) \) at \( x = 10 \) to find the local extreme value.

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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 the behavior of the function on different intervals.
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Critical Points

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if the derivative is negative. By analyzing the sign of the derivative on intervals between critical points, you can determine where the function is rising or falling, which is essential for identifying local extrema.
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Determining Where a Function is Increasing & Decreasing

Local Extrema

Local extrema are points where a function reaches a local maximum or minimum value. These occur at critical points where the derivative changes sign. To confirm whether a critical point is a local maximum or minimum, use the first or second derivative test, which examines the behavior of the derivative around these points.
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05:58
Finding Extrema Graphically
Pratica correlata
Domanda del libro di testo

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 / (x² - 1)

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Identifying Extrema


In Exercises 15–18:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local and absolute extreme values, if any, saying where they occur.


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Domanda del libro di testo

107. Marginal cost The accompanying graph shows the hypothetical cost c=f(x) of manufacturing x items. At approximately what production level does the marginal cost change from decreasing to increasing?

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Finding Functions from Derivatives


In Exercises 37–40, find the function with the given derivative whose graph passes through the point P.


f'(x) = 2x − 1, P(0,0)

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Domanda del libro di testo

Checking the Mean Value Theorem


Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.


f(x) =√(x − 1), [1, 3]

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Domanda del libro di testo

Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.

7. y=sin|x|, -2π≤x≤2π

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