STATIONARY POINT, TURNING POINT AND POINT OF INLFEXION

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 $\frac{dy}{dx}=\mathrm{0\; an}$d find the value of x.

iii. Find the corresponding value of y by substituting it into the equation $y=f\left(x\right)$. $\mathrm{}$$$

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

 

(Taken from example 3.1 in Engineering Mathematics II)

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