# Inflection Point

In Calculus, the graphs of the functions have curves. The direction of the curve defines the words concavity and inflection points. If the graph of a curve is bent upward, like an upright bowl or U shape, then the curve is said to be concave up or convex. If the graph of a curve is bent down, like an inverted U, then the curve is said to be concave down or simply concave.

For any function f(x), If f(x) and f'(x) are both differentiable functions, then f(x) is said to be concave up if f"(x) $\geq$ 0 and concave down if f"(x) $\leq$ 0. The point at which the graph changes its direction of concavity is called Inflection point and at this point f”(x) = 0.

**According to the inflection point theorem, **

"If f'(a)exists and f "(a) changes sign at x = a, then (a, f(a)) is an inflection point and if f"(a) exists then f"(a) = 0.This theorem states a necessary but not sufficient condition for x to be an inflection point for f. From the definition, it can also be stated that the sign of f'(x) must be the same on either side of the point (x, y). If the sign is positive, then the point is a rising point of inflection and if the sign is negative, then the point is a falling point of inflection.

Inflection points can also be categorized based on whether f'(x) is zero or not zero.

- If f'(x) = 0, then the point is called a stationary point of inflection, and is also called as a saddle-point.
- If f'(x) $\neq$ 0, then the point is a non-stationary point of inflection.

Look at the graph above. It represents the graph of y = $x^3$. At (0, 0) the tangent becomes the x-axis and cuts the graph at this point. This point (0, 0) is called the stationary point of inflection or saddle point.

Rotate the graph y = $x^3$ slightly about the origin (say about 20° in anticlockwise direction). At (0, 0) the tangent still cuts the graph in two, but its slope is non-zero. This is non-stationary point of inflection.

**In short, inflection point can be defined as:**

An inflection point on f(x) occurs at ‘a’ if and only if f(x) has a tangent line at ‘a’ and there exists an interval I containing ‘a’ such that f(x) is concave up on one side of ‘a’ and concave down on the other side.