FINDING CONCAVITY AND INFLECTION POINTS FOR A FUNCTION

What is concave up and down ?

Concave upward :

If y = f(x) is a concave upward curve, then as x increases f'(x) either is of the same sign and increasing or changes sign from negative to positive.

In either case f'(x) is increasing and so f''(x) > 0 for a concave upward curve f''(x) > 0.

Concave downward :

An arc of a curve y = f(x) is called concave downward, then as x increases, f'(x) either is the same sign and decreasing or changes sign from positive to negative.

In either case f'(x) is decreasing and so f''(x) < 0, for a concave downward curve f''(x) < 0

Procedure to find Concavity and Points of Inflection

To find concavity of a function y = f(x), we will follow the procedure given below.

Step 1 :

Find the first derivative f '(x). By equating the first derivative to 0, we will receive critical numbers.

Step 2 :

Find the second derivative, that is f ''(x). Equate the second derivative f ''(x) = 0.

Apply the values that we are deriving from the second derivative in the number line and decompose into intervals.

Step 3 :

From each interval, by applying one of the values in the second derivative, if we receive the second derivative.

• f''(x) > 0, then the function f(x) is concave up in the interval.
• f''(x) < 0, then the function f(x) is concave down in the interval.

Point of inflection :

A point of inflection is a point at which a curve is changing

concave upward to concave downward

or

concave downward to concave upward

A curve y = f(x) has one of its points x = c as an inflection point, if

i) f''(c) = 0 or is not defined and 

ii) f''(x) changes sign as x increases through x = c.

To find the point of inflection, we will follow the procedure given below.

Step 1 :

Equate the second derivative to 0, that is f ''(x) = 0

x = a, b

Step 2 :

(a, f(a)) and (b, f(b)) will be point of inflections.