Stationary Point, Turning Point and Point Of Inflexion

A, B and C are stationary points of the curve$y=f\left(x\right)$ . At stationary points, the gradient of tangent is 0 or $\frac{dy}{dx}=0$ . A and B are turning points. These are points where the curve changes direction. C is an inflexion point. An inflexion point is a point on a curve at which the sign of the curvature changes. At inflexion point $\frac{{d}^{2}y}{d{x}^{2}}=0$ but $\frac{dy}{dx}$ is not necessarily zero.

Turning point

At any turning point $\frac{dy}{dx}=0.$ Steps to determine the coordinates of the turning point of a curve:
i. Find $\frac{dy}{dx}$
ii. Let $\frac{dy}{dx}=0 an$d find the value of x.
iii. Find the corresponding value of y by substituting it into the equation $y=f\left(x\right)$ .

Example on how to find the co-ordinate of turning point of a curve.

(Taken from example 3.1 in Engineering Mathematics II)