Chapter 11, Problem 2CA

a.

To determine

To find: Whether there are any values of x greater than 0 where $f\left(x\right)>h\left(x\right)$ .

Answer to Problem 2CA

There are any values of x greater than 0 where $f\left(x\right)>h\left(x\right)$ .

Explanation of Solution

It is known that the functions $f\left(x\right)$ , $g\left(x\right)$ and $h\left(x\right)$ are linear, exponential and quadratic respectively.

Here, for the function $f\left(x\right)$ to be greater than $h\left(x\right)$ ; the number of points of intersection for $f\left(x\right)$ and $g\left(x\right)$ is two.

Hence, there are any values of x greater than 0 where $f\left(x\right)>h\left(x\right)$ .

b.

To determine

To find: Whether there are any values of x greater than 1 where $g\left(x\right)>f\left(x\right)$ .

Answer to Problem 2CA

There are any values of x greater than 1 where $g\left(x\right)>f\left(x\right)$ .

Explanation of Solution

It is known that the functions $f\left(x\right)$ , $g\left(x\right)$ and $h\left(x\right)$ are linear, exponential and quadratic respectively.

Here, for the function $g\left(x\right)$ to be greater than $f\left(x\right)$ ; the number of points of intersection for $f\left(x\right)$ and $g\left(x\right)$ is three.

Hence, there are any values of x greater than 1 where $g\left(x\right)>f\left(x\right)$ .

c.

To determine

To find: Whether there are any values of x greater than 0 where $g\left(x\right)>h\left(x\right)$ .

Answer to Problem 2CA

There are any values of x greater than 0 where $g\left(x\right)>h\left(x\right)$ .

Explanation of Solution

It is known that the functions $f\left(x\right)$ , $g\left(x\right)$ and $h\left(x\right)$ are linear, exponential and quadratic respectively.

Here, for the function $g\left(x\right)$ to be greater than $f\left(x\right)$ ; the function $g\left(x\right)$ increases without bound while the rate of change of the function $h\left(x\right)$ remains constant.

Hence, there are any values of x greater than 0 where $g\left(x\right)>h\left(x\right)$ .