Explanation Given Information: The provided function is, h ( x ) = 1 x + 1 x 2 Formula used: First Derivative Test: Suppose that c is a critical point of the continuous function f and its derivative is defined for x close to, and on both sides of x = c . Then, determine the sign of the derivative to the left and the right of x = c . If f ′ ( x ) is positive to the left of x = c and negative to the right, then f has a maximum at x = c . If f ′ ( x ) is negative to the left of x = c and positive to the right, then f has a minimum at x = c . If f ′ ( x ) has the same sign on both sides of x = c , then f has neither a maximum nor a minimum at x = c . Stationary Points are the points in the interior of the domain where the derivative is zero. To locate stationary points set f ′ ( x ) = 0 and solve for x . Singular Points are the points in the interior of the domain where the derivative is not defined. To locate singular points, find the interior points x where f ′ ( x ) is not defined but f ( x ) is defined. Closed intervals contain end points but open intervals do not have end points. Calculation: Consider the provided equation, h ( x ) = 1 x + 1 x 2 Differentiate the function h ( x ) = 1 x + 1 x 2 on both sides with respect to x as, h ′ ( x ) = − 1 x 2 − 2 x 3 Locate the singular points by f ′ ( x ) = 0 to get, − 1 x 2 − 2 x 3 = 0 Add, 1 x 2 to both sides of the equation to get, − 1 x 2 − 2 x 3 + 1 x 2 = 1 x 2 − 2 x 3 = 1 x 2 Cross multiplying to get, − 2 x 2 = x 3 Subtracting x 3 from both sides of the equation to get − 2 x 2 − x 3 = x 3 − x 3 − 2 x 2 − x 3 = 0 Multiply both sides by − 1 . − 2 x 2 − x 3 = 0 2 x 2 + x 3 = 0 Taking x 2 common from both sides of the equation to get x 2 ( 2 + x ) = 0 Thus, x 2 = 0 x = 0 Or 2 + x = 0 x = − 2 The equation h ′ ( x ) = − 1 x 2 − 2 x 3 is not defined at x = 0 and x = 0 is not an interior point of the domain. Thus, there are no singular points. Now, locate End points. Since, the domain is all real numbers except x = 0 . Thus, there are no end points. The equation has a stationary point at x = − 2

Student Solutions Manual for Waner/Costenoble's Finite Math and Applied Calculus, 7th

7th Edition

ISBN: 9781337275972

Author: Waner, Stefan; Costenoble, Steven

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Videos

Question

Chapter 12, Problem 7RE

To determine

To calculate: The exact location of the relative and absolute extrema of the function h(x)=1x+1x2.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 12, Problem 6RE

Chapter 12, Problem 8RE

Students have asked these similar questions

4. Find the inverse Laplace Transform Show all of your work: a. F(s) = = 2s-3 (s²-10s+61)(5-3) se-2s b. G(s) = (s+2)²

1. Consider the differential equation, show all of your work: dy =(y2)(y+1) dx a. Determine the equilibrium solutions for the differential equation. b. Where is the differential equation increasing or decreasing? c. Where are the changes in concavity? d. Suppose that y(0)=0, what is the value of y as t goes to infinity?

2. Suppose a LC circuit has the following differential equation: q'+4q=6etcos 4t, q(0) = 1 a. Find the function for q(t), use any method that we have studied in the course. b. What is the transient and the steady-state of the circuit?

Chapter 12 Solutions

Student Solutions Manual for Waner/Costenoble's Finite Math and Applied Calculus, 7th

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The exact location of the relative and absolute extrema of the function h ( x ) = 1 x + 1 x 2 .

Solution Summary: The author explains how to calculate the exact location of relative and absolute extrema of the function h(x)=1x

Concept explainers

Videos

To calculate: The exact location of the relative and absolute extrema of the function h(x)=1x+1x2.

Want to see the full answer?

Chapter 12 Solutions