Book edition 2nd

Author(s) James Stewart

Pages 830 pages

ISBN 9781133112280

Answers without the blur.

Just sign up for free and you're in.

Short Answer

(a) Use a graph of \[f\] to estimate the maximum and minimum values. Then find the exact values.
(b) Estimate the value of \[x\] at which \[f\] increases most rapidly. Then find the exact value.

\[f\left( x \right) = {x^2}{e^{ - x}}\]

(a) The local and local and abs max lies at \(\left( {2,4{e^{ - 2}}} \right)\), \(\left( {0,0} \right)\).

(b) The resultant answer is \(x = 2 - \sqrt 2 \).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given data

The given function is \(x{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{and}}{\kern 1pt} {\kern 1pt} {\kern 1pt} f\).

Step 2: Concept of Differentiation

Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.

(a)Step 3: Sketch the graph

The graph is shown below:

\(\begin{array}{l}x \approx 2\\x \approx 0\end{array}\)

Step 4: Determine the value of the function

Determine the value of the function \(f(x)\):

\(\begin{array}{l}f(x) = {x^2}{e^{ - x}}\\f\prime (x) = 2x{e^{ - x}} - {x^2}{e^{ - x}}\\f\prime (x) = x{e^{ - x}}(2 - x)\end{array}\)

Similarly, calculate further:

\(\begin{array}{l}x = 2\\x = 0\end{array}\)

(b) Step 5: Simplify the expression

The problem of finding exact value of \(x\) at which \(f\) increases most rapidly is the same as finding value of \(x\), such that \({f^\prime }(x)\) is maximum.

\(f(x) = {x^2}{e^{ - x}}\), Therefore,\({f^\prime }(x) = 2x{e^{ - x}} - {x^2}{e^{ - x}}\)

To find value of \(x\), such that \({f^\prime }(x)\) is maximum, we equate \({f^{\prime \prime }}(x) = 0\)

\({f^{\prime \prime }}(x) = 2{e^{ - x}} - 2x{e^{ - x}} - 2x{e^{ - x}} + {x^2}{e^{ - x}}\)

Simplify further;

\(\begin{array}{l}{f^{\prime \prime }}(x) = \left( {2 - 2x - 2x + {x^2}} \right){e^{ - x}}\\{f^{\prime \prime }}(x) = {e^{ - x}}\left( {{x^2} - 4x + 2} \right)\end{array}\)

to have \({f^{\prime \prime }}(x) = 0\) is same as to have \({x^2} - 4x + 2 = 0\), since, \({e^{ - x}}\) cannot be 0

Therefore, \(x = 2 \pm \sqrt 2 ,x = 2 - \sqrt 2 \) is the answer and not \(x = 2 + \sqrt 2 \), because for the former value of \(x,{f^\prime }(x)\) is increasing before and decreasing after the point in the neighborhood, therefore a maximum.

Find Study Materials for

Create Study Materials

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

(a) Use a graph of \[f\] to estimate the maximum and minimum values. Then find the exact values.
(b) Estimate the value of \[x\] at which \[f\] increases most rapidly. Then find the exact value.

\[f\left( x \right) = {x^2}{e^{ - x}}\]

Step by Step Solution

Step 1: Given data

Step 2: Concept of Differentiation

(a)Step 3: Sketch the graph

Step 4: Determine the value of the function

(b) Step 5: Simplify the expression

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Essential Calculus: Early Transcendentals

Answers without the blur.

Short Answer

(a) Use a graph of \[f\] to estimate the maximum and minimum values. Then find the exact values. (b) Estimate the value of \[x\] at which \[f\] increases most rapidly. Then find the exact value. \[f\left( x \right) = {x^2}{e^{ - x}}\]

Step by Step Solution

Step 1: Given data

Step 2: Concept of Differentiation

(a)Step 3: Sketch the graph

Step 4: Determine the value of the function

(b) Step 5: Simplify the expression

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

(a) Use a graph of \[f\] to estimate the maximum and minimum values. Then find the exact values.
(b) Estimate the value of \[x\] at which \[f\] increases most rapidly. Then find the exact value.

\[f\left( x \right) = {x^2}{e^{ - x}}\]