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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

Investigate the family of functions f ( x ) = c x e c x 2 . What happens to the maximum and minimum points and the inflection points as c changes? Illustrate your conclusions by graphing several members of the family.

To determine

To investigate: For different values of c the behavior of the function f(x)=cxecx2 over the interval (,) .

Explanation

The given function is f(x)=cxecx2 .

When c=0 the function is f(x)=0

When c=1 the function is f(x)=xex2

When c=2 the function is f(x)=2xe2x2

When c=3 the function is f(x)=3xe3x2

Draw the graph of above function as shown in below Figure 1

To examine local minima or maxima differentiate the function f with respect to x.

f(x)=cxecx2f(x)=c(x(2cx(ecx2))+ecx2)=2c2x2ecx2+cecx2=cecx2(12cx2)

Set f(x)=0 and find the value of x

f(x)=0cecx2(12cx2)=012cx2=0x=±12c

This is the critical point for f(x) .

Substitute the value of x in the f(x)

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