Textbook Question
110. Suppose the derivative of the function y = f(x) is
y'=(x-1)^22(x-2)(x-4).
At what points, if any, does the graph of f have a local minimum, local maximum, or
point of inflection?
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 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)
Verified step by step guidance1
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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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 the behavior of the function on different intervals.
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04:50
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.
Recommended video:
07:32Determining 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.
Recommended video:
05:58Finding Extrema Graphically
Related Practice
Textbook Question
Absolute Extrema on Finite Closed Intervals
In Exercises 21–36, find the absolute maximum and minimum values of each function on the given interval. Then graph the function. Identify the points on the graph where the absolute extrema occur, and include their coordinates.
g(x) = √(4 − x²), −2 ≤ x ≤ 1
Textbook Question
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.
Textbook Question
26. Constructing cylinders Compare the answers to the following two construction problems.
a. A rectangular sheet of perimeter 36 cm and dimensions x cm by y cm is to be rolled into a cylinder as shown in part (a) of the figure. What values of x and y give the largest volume?
b. The same sheet is to be revolved about one of the sides of length y to sweep out the cylinder as shown in part (b) of the figure. What values of x and y give the largest volume?
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Textbook Question
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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Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
g(t) = √t + √(1 + t) − 4, (0, ∞)