Chapter 2, Problem 16RCC

(a)

To determine

To find:The significance of the sign $f\text{'}\left(x\right)$ over $f\left(x\right)$ .

Answer to Problem 16RCC

The significance of the sign $f\text{'}\left(x\right)$ over $f\left(x\right)$ is When $f\text{'}\left(x\right)>0$ the function $f\left(x\right)$ is increasing and for $f\text{'}\left(x\right)<0$ ,the function $f\left(x\right)$ is decreases.

Explanation of Solution

Given information:

The given statement is “The significance of the sign $f\text{'}\left(x\right)$ over $f\left(x\right)$ ”.

Calculation:

The derivative $f\text{'}\left(x\right)$ represent the slope of the curve $y=f\left(x\right)$ at the point $\left(x,f\left(x\right)\right)$ .

When $f\text{'}\left(x\right)>0$ , the slope is positive and hence the function $f\left(x\right)$ is increasing on that interval.

When $f\text{'}\left(x\right)<0$ , the slope is positive and hence the function $f\left(x\right)$ is decreases on that interval.

Therefore, the significance of the sign $f\text{'}\left(x\right)$ over $f\left(x\right)$ is When $f\text{'}\left(x\right)>0$ the function $f\left(x\right)$ is increasing and for $f\text{'}\left(x\right)<0$ ,the function $f\left(x\right)$ is decreases.

(b)

To determine

To find:The significance of the sign $f\text{'}\text{'}\left(x\right)$ over $f\left(x\right)$ .

Answer to Problem 16RCC

The significance of the sign $f\text{'}\text{'}\left(x\right)$ over $f\left(x\right)$ is When $f\text{'}\text{'}\left(x\right)>0$ the function $f\left(x\right)$ is concave upward and for $f\text{'}\text{'}\left(x\right)<0$ ,the function $f\left(x\right)$ is concave downward.

Explanation of Solution

Given information:

The given statement is “The significance of the sign $f\text{'}\text{'}\left(x\right)$ over $f\left(x\right)$ ”.

Calculation:

  $f\text{'}\text{'}\left(x\right)=\left(f\text{'}\left(x\right)\right)\text{'}$ , if is positive, then $f\text{'}\left(x\right)$ is an increasing function. Then the slope of the tangent lines of the curve $y=f\left(x\right)$ increase from left to right.

If the value of $x$ increases the slope of the curve becomes progressively larger and the curve bends upward. Such a curve called concave upward.

Therefore, if $f\text{'}\text{'}\left(x\right)$ is negative, then $f\text{'}\left(x\right)$ is decreasing.

the slope of $f\left(x\right)$ decreases from left to right and the curve bends downward. This curve is called concave downward.

From the above facts.

If $f\text{'}\text{'}\left(x\right)>0$ on an interval, then $f\left(x\right)$ is concave upward on that interval.

If $f\text{'}\text{'}\left(x\right)<0$ on an interval, then $f\left(x\right)$ is concave downward on that interval.

Therefore, the significance of the sign $f\text{'}\text{'}\left(x\right)$ over $f\left(x\right)$ is When $f\text{'}\text{'}\left(x\right)>0$ the function $f\left(x\right)$ is concave upward and for $f\text{'}\text{'}\left(x\right)<0$ ,the function $f\left(x\right)$ is concave downward.