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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.2.21
Chapter 4, Problem 4.2.21

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, ∞)

Verified step by step guidance
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First, understand that we need to show the function g(t) = √t + √(1 + t) − 4 has exactly one zero in the interval (0, ∞). This means we need to find where g(t) = 0 and prove it happens only once in the given interval.
To start, evaluate the behavior of g(t) as t approaches 0 and as t approaches infinity. As t approaches 0, g(t) approaches √0 + √1 - 4 = 1 - 4 = -3. As t approaches infinity, both √t and √(1 + t) grow without bound, so g(t) approaches infinity.
Next, check the continuity and differentiability of g(t) in the interval (0, ∞). Since both √t and √(1 + t) are continuous and differentiable for t > 0, g(t) is also continuous and differentiable in (0, ∞).
Apply the Intermediate Value Theorem. Since g(t) is continuous on (0, ∞) and changes from negative to positive as t increases from 0 to infinity, there must be at least one zero in the interval (0, ∞).
To show there is exactly one zero, consider the derivative g'(t) = (1/2√t) + (1/2√(1 + t)). Since both terms are positive for t > 0, g'(t) > 0 for all t in (0, ∞), indicating that g(t) is strictly increasing. Therefore, g(t) can cross the x-axis at most once, confirming exactly one zero in the interval.

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Key Concepts

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

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function, f, takes on values of opposite sign at two points, a and b, then it must have at least one root in the interval (a, b). This concept is crucial for proving the existence of a zero in a given interval by showing that the function changes sign.
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Fundamental Theorem of Calculus Part 1

Continuity of Functions

A function is continuous on an interval if it is continuous at every point within that interval. For the function g(t) = √t + √(1 + t) − 4, continuity is essential to apply the Intermediate Value Theorem. The square root functions involved are continuous for t > 0, ensuring g(t) is continuous on (0, ∞).
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Intro to Continuity

Behavior of Functions at Infinity

Understanding how a function behaves as t approaches infinity helps determine the number of zeros. For g(t) = √t + √(1 + t) − 4, as t increases, both √t and √(1 + t) grow, suggesting g(t) will eventually become positive. This behavior, combined with initial negative values, supports the existence of exactly one zero.
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Graphs of Exponential Functions
Related Practice
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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Textbook Question

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = {x² − x, −2 ≤ x ≤−1

2x² − 3x − 3, −1 < x ≤ 0

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Textbook Question

56. Airplane landing path An airplane is flying at altitude H when it begins its descent to an airport runway that is at horizontal ground distance L from the airplane, as shown in the accompanying figure. Assume that the landing path of the airplane is the graph of a cubic polynomial function y = ax^3+bx^2+cx+d, where y(-L)= H and y(0)=0.

a. What is dy/dx at x = 0?

b. What is dy/dx at x = -L?

c. Use the values for dy/dx at x = 0 and x =- L together with y(0) = 0 and y(-L) = H to show that y(x)=H[2(x/L)^3+3(x/L)^2]

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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 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)

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.