Question

(a) State the Increasing/Decreasing Test.

(b) What does it mean to say that is concave upward on an interval.

c) State the Concavity Test.

d) what are inflection points? How do you find them?

Short answer

(a) The increasing/decreasing test is stated.

(b) A function is said to be concave upward on an interval \(I\), if \(f''\left( x \right) > 0\) at each point in the interval \(I\).

(c) The concavity test is stated.

(d) The smooth turning point of a plane where the mark changes curve is called the point of inflection, rotation point, flex, or inflection. The steps to find inflection points are stated.

Step-by-step solution

Part (a)

The function \(f\left( x \right)\) is increase periodically if the in \(x\) values increase, means,\(y\) value increase for all\(x\)values, since

\(\begin{array}{c}{x_1} > {x_2}\\ = f\left( {{x_1}} \right) < f\left( {{x_2}} \right)\end{array}\)

And, \(f\left( x \right)\) is decreasing if,

\(\begin{array}{c}{x_1} < {x_2}\\ = f\left( {{x_1}} \right) > f\left( {{x_2}} \right)\end{array}\)

The geometrical representation of increasing/decreasing function \(f\left( x \right)\) is,

Part (b)

A function is said to be concave upward on an interval\(I\). If \(f''\left( x \right) > 0\) at each point in the interval \(I\).

• The function is concave upward while the slope keeps rising.
• If the second output is positive, the function becomes concave at the top.

The graph for such functions is,

Concavity test (c)

let\(f\)be twice differentiable on an interval\(I\).

The graph of f is concave up if \(f'' > 0\) on \(I\),

And is concave down if \(f'' < 0\) on \(I\).

Graph

The four ways concavity is linked to increase/decrease, as well as the relationship with the output of the first and second ones.

In terms of geometry, the function turns when the graph is above its tangent lines. The work is concave down when its graph is below its tangent lines.