Chapter 5.4, Problem 28E

a.

To determine

To find the absolute maximum of $f.$

Answer to Problem 28E

Absolute max i.e., other are no other extrema.

Explanation of Solution

Given Derivative $f^{\prime }=e^{-x}+x\left(-e^{-x}\right).$

Calculation:

Derivative $f^{\prime }=e^{-x}+x\left(-e^{-x}\right)$ ,roots at $e^{-x}-x\left(-e^{-x}\right)=0,x=1\text{ and }f\left(1\right)=1=\text{local max}$ .

  $f^{\prime }$ is positive for $x<1$ and negative for $x>1$ ; then $x=1$ Absolute max i.e. ,other are no other extrema.

b.

To determine

To find the area of the rectangle.

Answer to Problem 28E

The area of the rectangle is $A=\left(b-a\right)x{e}^{-x}.$

Explanation of Solution

Given rectangle.

Calculation:

Here $b$ can be computed by solve for $x$ , which is very difficult, or can use a “graph utility”

The area is just the width time’s height.

  $A=\left(b-a\right)x{e}^{-x}$

  

c.

To determine

To draw a scatter plot of the data $\left(a,A\right)$ .

Answer to Problem 28E

  

Explanation of Solution

Given Scatter plot.

Calculation:

Scatter plot of the $\left(a,A\right)$ pairs from part (b).

  

d.

To determine

To find the quadratic, cubic, and quadratic regression equations.

Answer to Problem 28E

The required equations are, (Red) quadratic: $A=-0.901515a^2+0.514697a+\mathrm{0.345.}$

(Purple) cubic: $A=1.91142a^3-4.05536a^2+1.96929a+\mathrm{0.181.}$

(Green) quadratic: $A=-1.77739a^4+5.82168a^3-6.91696a^2+2.75134a+\mathrm{0.12.}$

Explanation of Solution

Given Scatter plot.

Calculation:

  

From the $\left(a,A\right)$ data of part (b), the regression capabilities of “some graph utility” produce:

(Red) quadratic: $A=-0.901515a^2+0.514697a+\mathrm{0.345.}$

(Purple) cubic: $A=1.91142a^3-4.05536a^2+1.96929a+\mathrm{0.181.}$

(Green) quadratic: $A=-1.77739a^4+5.82168a^3-6.91696a^2+2.75134a+\mathrm{0.12.}$

e.

To determine

To find the maximum possible value of area of the rectangle.

Answer to Problem 28E

The maximum possible value of the area of the rectangle $A=\mathrm{0.47.}$

Explanation of Solution

Given Scatter plot.

Calculation:

  

From the $\left(a,A\right)$ data of part (b),the regression capabilities of “some graph utility” produces:

(Red) quadratic: $A=-0.901515a^2+0.514697a+\mathrm{0.345.}$

(Purple) cubic: $A=1.91142a^3-4.05536a^2+1.96929a+\mathrm{0.181.}$

(Green) quadratic: $A=-1.77739a^4+5.82168a^3-6.91696a^2+2.75134a+\mathrm{0.12.}$

The quadratic one is not too close of a fit so it will not really consider it. the quartic curve follows the data the best and its maximum roughly coincides with $a=0.3$ .from the table in part (b) , $A=\mathrm{0.47.}$