Explanation Definition used: First derivative test: Suppose c to be a critical number for a function f . Assume that f is continuous on ( a , b ) and differentiable on ( a , b ) except at c , and c is the only critical number of f in ( a , b ) . f ( c ) is a relative maximum of f at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . f ( c ) is a relative minimum f at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Relative maximum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) > 0 in the interval ( a , c ) and f ′ ( x ) < 0 in the interval ( c , b ) . Relative minimum: A function is said to have a relative maximum f ( c ) at c if f ′ ( x ) < 0 in the interval ( a , c ) and f ′ ( x ) > 0 in the interval ( c , b ) . Calculation: The given function is f ( x ) = x e x x − 1 . Obtain the derivative of the function as follows. f ′ ( x ) = ( x e x x − 1 ) ′ = ( x − 1 ) ( x e x ) ′ − ( x e x ) ( x − 1 ) ′ ( x − 1 ) 2 = ( x − 1 ) ( x e x + e x ) − ( x e x ) ⋅ 1 ( x − 1 ) 2 = x 2 e x + x e x − x e x − e x − x e x ( x − 1 ) 2 Simplify further as follows. f ′ ( x ) = x 2 e x + x e x − x e x − e x − x e x ( x − 1 ) 2 = x 2 e x − e x − x e x ( x − 1 ) 2 = e x ( x 2 − 1 − x ) ( x − 1 ) 2 Clearly, at x = 1 , the derivative does not exist but it cannot be a critical number as the function is not defined at this point. Also, the derivative does not exist at a point c that is undefined. Set f ′ ( c ) = 0 and find the value of c . e c ( c 2 − 1 − c ) ( c − 1 ) 2 = 0 e c ( c 2 − c − 1 ) = 0 ( c 2 − c − 1 ) = 0 Compute the value of c with the help of quadratic formula as follows. c = 1 ± 1 2 − 4 × 1 × ( − 1 ) 2 × 1 = 1 ± 5 2 = 1 ± 2.2361 2 Simplify further as follows. c = 1 ± 2.2361 2 c = 1 + 2

Finite Mathematics and Calculus with Applications (10th Edition)

10th Edition

ISBN: 9780321979407

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

See similar textbooks

Concept explainers

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …

Videos

Question

Chapter 13, Problem 31RE

To determine

To find: The x-values of all points where the function f(x)=xexx−1 has relative maxima and minima, and to find the values of all relative extrema.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 13, Problem 30RE

Chapter 13, Problem 32RE

Students have asked these similar questions

Determine the smallest value of the constant a for which the graph of the function f(x) = a^x−xis always above the x−axis.

What x value maximizes y in the function y^2 +y+x^2-2x=5

Find the relative maxima and relative minima, if any, of the function. (If an answer does not exist, enter DNE.) F(t) = 3t5 − 5t3 + 12 relative maximum (x,y)= relative minimum (x,y)=

Chapter 13 Solutions

Finite Mathematics and Calculus with Applications (10th Edition)

Ch. 13.1 - Find where the function is increasing and...Ch. 13.1 - Prob. 2YT Ch. 13.1 - Prob. 3YT Ch. 13.1 - Prob. 4YT Ch. 13.1 - Prob. 1WE Ch. 13.1 - Prob. 2WE Ch. 13.1 - Prob. 3WE Ch. 13.1 - Prob. 4WE Ch. 13.1 - Prob. 5WE Ch. 13.1 - Prob. 6WE

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 x -values of all points where the function f ( x ) = x e x x − 1 has relative maxima and minima , and to find the values of all relative extrema.

Solution Summary: The author explains how to find the x -values of all points where the function f(x)=x

Concept explainers

Videos

To find: The x-values of all points where the function f(x)=xexx−1 has relative maxima and minima, and to find the values of all relative extrema.

Want to see the full answer?

Chapter 13 Solutions