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

7th Edition

ISBN: 9781337291248

Author: Waner, Stefan.

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 5, 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 5, Problem 6RE

Chapter 5, Problem 8RE

Students have asked these similar questions

A manufacturer of handcrafted wine racks has determined that the cost to produce x units per month is given by C = 0.3x² + 7,000. How fast is the cost per month changing when production is changing at the rate of 14 units per month and the production level is 80 units? Costs are increasing at the rate of $ (Round to the nearest dollar as needed.) per month at this production level.

dy Assume x and y are functions of t. Evaluate for 2xy -3x+2y³ = - 72, with the conditions dt dx dt = -8, x=2, y = -3. dy dt (Type an exact answer in simplified form.)

Consider the sequence below: 1 1 1 (a) Express this sequence as a recurrence relation (b) Express this sequence in the form {a}=1 (c) Does this sequence converge or diverge? Justify your answer. Consider the sequence below: 1 1 1 1, 4' 9' 16' (a) Express this sequence in the form {ak}=1 (b) Does this sequence converge or diverge? Justify your answer. Consider the sequence below: 345 2. 4' 9' 16' ·} (a) Express this sequence in the form {a}1 (b) Does this sequence converge or diverge? Justify your answer.

Chapter 5 Solutions

Applied Calculus

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 5 Solutions