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?
1
views
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 4.2.21
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 guidance1
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mKey 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.
Recommended video:
Guided course
06:11
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, ∞).
Recommended video:
05:34Intro 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.
Recommended video:
5:46Graphs of Exponential Functions
Related Practice
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]
1
views
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.
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.
g(x) = x√8 − x²