Textbook Question
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ ― 2𝓍 + 4
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.1.45
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x² + 2/x
Verified step by step guidance1
First, identify the domain of the function y = x² + 2/x. Since the function includes a term 2/x, x cannot be zero. Therefore, the domain is all real numbers except x = 0.
To find the critical points, we need to find the derivative of the function y = x² + 2/x. Use the power rule and the quotient rule to differentiate: y' = d/dx (x²) + d/dx (2/x).
Calculate the derivative: y' = 2x - 2/x². This involves differentiating x² to get 2x and using the power rule on 2/x to get -2/x².
Set the derivative equal to zero to find critical points: 2x - 2/x² = 0. Solve this equation for x to find the values where the slope of the tangent is zero.
Additionally, check the endpoints of the domain. Since the domain is all real numbers except x = 0, consider the behavior of the function as x approaches zero from both the positive and negative sides to understand the behavior at the domain boundaries.
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4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Critical Points
Critical points of a function occur where its derivative is zero or undefined. These points are important because they can indicate local maxima, minima, or points of inflection. To find them, take the derivative of the function and solve for the values of x where the derivative equals zero or does not exist.
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Critical Points
Derivative
The derivative of a function represents the rate of change of the function with respect to its variable. It is a fundamental tool in calculus used to find slopes of tangent lines, velocities, and other rates of change. For the function y = x² + 2/x, the derivative is found using the power rule and the quotient rule.
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05:44Derivatives
Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions like y = x² + 2/x, the domain excludes values that make the denominator zero. Identifying the domain is crucial for understanding where critical points and endpoints can occur.
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Related Practice
Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x³ + 4x² + 7, (−∞, 0)
Textbook Question
Applications
Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?
Textbook Question
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.
53. y = x * √(8 - x²)
Textbook Question
54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)
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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.