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Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

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Section
BuyFindarrow_forward

Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

In Problems 39-42, find any relative maxima and minima. Use a graphing utility to check your results.

y = e x x

To determine

To calculate: The relative maxima or minima for the function, y=exx.

Explanation

Given Information:

The function is y=exx.

Formula used:

If f(x)=cu(x), where, c is a constant and u(x) is a differentiable function of x, then,

f(x)=cu(x)

According to the property of derivatives, if y=ex, where, u is a differentiable function of x,

dydx=ex

According to the power rule of differentiation,

dydx=nxn1

According to the product rule, if a function is of the form y=fg, then the derivative of the function is

y=gf+fg

According to the quotient rule, if a function is of the form y=fg, then the derivative of the function is

y=gffgg2

Calculation:

Consider the provided function,

y=exx

To find the relative minimum, differentiate both sides with respect to x,

dydx=ddx(exx)=ddx(exx)

Apply the quotient rule for derivatives,

dydx=x(ddx(ex))(ex)(ddx(x))x2

Simplify the derivative using the quotient rule of derivatives,

dydx=x(ex)(ex)(x11)x2=xexexx2

Take ex common from the numerator,

dydx=ex(x1)x2

To calculate the value of mean, the value of dydz should be equal to zero.

dydz=0ex(x1)x2=0x1=0x=1

To find whether the function has relative minima or maxima, find the derivative of dydx,

d2ydx2=ddx(ex(x1)x2)

Simplify the derivative using the quotient rule for derivatives,

d2ydx2=x2(ddx(xexex))(xexex)(ddx(x

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Chapter 11 Solutions

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