Chapter 4, Problem 5RCC

(a)

To determine

To state: The increasing / decreasing test.

Explanation of Solution

The Increasing/decreasing test:

1) If $f^{\prime }\left(x\right)>0$ on an interval then $f$ is increasing on that interval.

2) If $f^{\prime }\left(x\right)<0$ on an interval then $f$ is decreasing on that interval.

(b)

To determine

To explain: The meaning of concave upward curve on an interval.

Explanation of Solution

If graph of a function f lies above all its tangents on interval I then f is known as concave upward over interval I.

In Figure 1, it is shown that for a concave upward curve the tangent lies below the graph of the curve while a concave downward curve the tangent lies above the graph of the curve.

(c)

To determine

To state: The concavity test.

Explanation of Solution

The concavity test:

1) If ${f^{\prime }}^{\prime }\left(x\right)>0$ for all x in I then graph of f is concave upward on interval I.

2) If ${f^{\prime }}^{\prime }\left(x\right)<0$ for all x in I then graph of f is concave downward on interval I.

(d)

To determine

To define: The inflections points and explain the method of finding them.

Explanation of Solution

Inflection point:

“A point P on a curve $y=f\left(x\right)$ is called inflection point if f is continuous at P and the curve changes from concave upward to concave downward or vice versa”.

An inflection point is at $\left(x,f\left(x\right)\right)$ for any value of x where the concavity changes.