Concave, convex functions and points of inflection

 

Concave functions

If f''(x) ≤ 0 in a given interval of x, the function is said to be concave.

Convex functions

If f''(x) ≥ 0 in a given interval of x, the function is said to be convex.

Point of Inflection

The point at which a curve changes being concave to convex or vice versa is called a point of inflection.

E.g.

f(x) = x³ - 2x² + 5x - 4

f'(x) = 3x² - 4x + 5

f''(x) = 6x - 4

At the point of inflection,

f''(x) = 0

6x - 4 = 0

x = 2/3 = 0.67

There is a point of inflection at x = 0.67.

If x = 0.6

f''(0.6) = 3.6 - 4 = -0.4

f''(0.6) < 0 - the function is concave.

If x = 0.8

f''(0.8) = 4.8 - 4 = 0.8

f''(0.8) > 0 - the function is convex.

In the above simulation, there are two points of inflection. The simulation can be practised interactively here; just move the point gently to see the change.

 

Points of inflection on the Bell Curve - Normal Distribution

There are two points of inflection on the bell curve. You can practise the following simulation interactively to understand it.