   Chapter 4.3, Problem 60E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (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(x) = x2e−x

(a)

To determine

To estimate: The maximum and minimum values of the given function by using online graphing calculator and also find the exact value.

Explanation

Given:

The function is f(x)=x2ex .

Calculation:

Use the online graphing calculator to obtain the graph of the given function f which is shown below in Figure 1.

From Figure 1, it is observed that there is no maximum value because the curve tends to positive infinity and also f has local maximum at x=2 . Therefore, the maximum value is f(2)0.5

Also from Figure 1, it is observed that the curve never came below the x-axis and it touches 0 only once. So, the function f has local and absolute minimum both at x=0 . Therefore the maximum value is f(0)0

To find the exact value obtain the derivative of f(x) .

f(x)=ddx(x2ex)

Apply the Product Rule.

f(x)=x2(ex)+ex(x2)=x2(ex)+ex(2x)=xe2(x+2)

Set f(x)=0 to find the critical point

(b)

To determine

To estimate: The value of x when f increases most quickly and also find the exact value.

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