Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.2.23

Chapter 4, Problem 4.2.23

"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero

Verified step by step guidance

Identify the function given in the exercise. Let's denote it as f(x).

Determine the domain of the function f(x) to ensure it is defined for all x in the interval of interest.

Use the Intermediate Value Theorem (IVT) to show that there is at least one zero. The IVT states that if a function is continuous on a closed interval [a, b] and f(a) and f(b) have opposite signs, then there is at least one c in (a, b) such that f(c) = 0.

To show that there is exactly one zero, demonstrate that the function is either strictly increasing or strictly decreasing on its domain. This can be done by finding the derivative f'(x) and showing that it does not change sign.

Conclude that since the function is continuous and either strictly increasing or decreasing, it can cross the x-axis at most once, confirming that there is exactly one zero.

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Key Concepts

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

Roots of a Function

The roots or zeros of a function are the values of the variable that make the function equal to zero. Finding the roots involves solving the equation f(x) = 0. This concept is crucial for understanding where the graph of the function intersects the x-axis, indicating the points where the function has no value.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function, f(x), takes on different signs at two points, then it must cross zero at some point between them. This theorem is essential for proving the existence of a zero within an interval, especially when the function changes from positive to negative or vice versa.

Uniqueness of Zeros

To show that a function has exactly one zero, it is important to demonstrate that the function is either strictly increasing or decreasing, ensuring no other zeros exist. This involves analyzing the derivative of the function to confirm its monotonic behavior, which guarantees that the function crosses the x-axis only once.

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Textbook Question

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

222

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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) = x³ / (3x² + 1)

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Textbook Question

Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

y = x² + 2/x

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Textbook Question

Absolute Extrema on Finite Closed Intervals

In Exercises 37–40, find the function’s absolute maximum and minimum values and say where they occur.

f(x) = x⁴ᐟ³, −1 ≤ x ≤ 8

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Textbook Question

Solve the initial value problems in Exercises 71–90.

y⁽⁴⁾ = −sin t + cos t;

y′′′(0) =7, y′′(0) = y′(0) = −1, y(0) = 0

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Textbook Question

Checking the Mean Value Theorem

Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.

g(x) = {x³, −2 ≤ x ≤ 0

x², 0 < x ≤ 2

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