# Inflection Points (Points of Inflection)

## Horizontal (stationary) point of inflection (inflection point)

If $x \lt a$, then $f'(x) \gt 0$.
If $x = a$, then $f'(x) = 0$ and $f''(x) = 0$.
If $x \gt a$, then $f'(x) \gt 0$.
That is, the gradient is positive either side of the stationary point.

If $x \lt a$, then $f'(x) \lt 0$.
If $x = a$, then $f'(x) = 0$ and $f''(x)=0$.
If $x \gt a$, then $f'(x) \lt 0$.
That is, the gradient is negative either side of the stationary point.

## Not all points of inflection (inflection points) are stationary points

The gradient of the tangent is not equal to 0. At the point of inflection, $f'(x) \ne 0$ and $f''(x)=0$.
When determining the nature of stationary points it is helpful to complete a 'gradient table', which shows the sign of the gradient either side of any stationary points. This is known as the first derivative test.

### Example 1

Find the points of inflection (inflection points) of $f(x)=2x^3-18x^2+30x+1$.

### Example 2

Find the points of inflection (inflection points) of $f(x)=x^3+6x^2+12x+7$.