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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.1.77a
Chapter 4, Problem 4.1.77a

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The function f(x) = √x has a local maximum on the interval [0,∞).

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To determine if the function f(x) = √x has a local maximum on the interval [0, ∞), we first need to understand what a local maximum is. A function f(x) has a local maximum at a point x = c if there exists an interval (a, b) containing c such that f(c) ≥ f(x) for all x in (a, b).
Next, consider the behavior of the function f(x) = √x. As x increases from 0 to ∞, the value of √x also increases. This suggests that the function is monotonically increasing on the interval [0, ∞).
To further analyze, we can take the derivative of f(x) to examine its critical points. The derivative f'(x) = (1/2)x^(-1/2) is positive for all x > 0, indicating that the function is increasing on its entire domain.
Since the function is increasing and has no points where the derivative is zero or undefined (other than at x = 0, where the derivative is undefined but the function is still increasing), there are no local maxima on the interval [0, ∞).
Therefore, the statement that the function f(x) = √x has a local maximum on the interval [0, ∞) is false. The function does not have a local maximum because it continuously increases without reaching a peak.

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Key Concepts

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

Local Maximum

A local maximum of a function occurs at a point where the function's value is greater than the values of the function at nearby points. To determine if a function has a local maximum, one typically examines the first derivative to find critical points and the second derivative to assess concavity. In the context of the given function, understanding local maxima is essential to evaluate its behavior over the specified interval.
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The Second Derivative Test: Finding Local Extrema

Derivative and Critical Points

The derivative of a function provides information about its rate of change and is used to find critical points, where the derivative is zero or undefined. These points are potential candidates for local maxima or minima. For the function f(x) = √x, calculating the derivative helps identify whether there are any points in the interval [0, ∞) where the function reaches a local maximum.
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Critical Points

Behavior of the Function on the Interval

Analyzing the behavior of the function on the specified interval is crucial for determining the presence of local maxima. For f(x) = √x, one must consider how the function behaves as x approaches 0 and as x increases towards infinity. Understanding the overall trend of the function helps in concluding whether it can attain a local maximum within the given interval.
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Graphs of Exponential Functions
Related Practice
Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.

Textbook Question

Optimal soda can


a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.

Textbook Question

Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)


a. How many people should the guide take on a tour to maximize the profit?

Textbook Question

Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.


a. Evaluate g(2), h(2), g'(2), and h'(2).

Textbook Question

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>

b. Find the time and the displacement when the object reaches its lowest point.

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Textbook Question

{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.


a. Plot a graph of the curve when a = 3.