If  $\frac{\text{d}x}{\text{d}y}=\infty$  , then the tangent at a point P(x,y) on the curve is perpendicular to the x- axis. 

The function f(x) is said to have attained its maximum value for x=a if the function ceases to increase and begins to decrease at x= a

The function is said to have attained its minimum value for x=b if the function ceases to decrease and begins to increase at x=b

The maximum or minimum point are called the turning or stationary point.

For maximum point (i)

For minimum point (i)  $\frac{\text{d}x}{\text{d}y}=0$  
(ii)  $\frac{\text{d}x}{\text{d}y}$  changes sign from -ve to +ve 

For Maximum point : (i)  $\frac{\text{d}x}{\text{d}y}=0$ ; (ii)  $\frac{d^2y}{dx^2}$  < 0

For minimum point : (i)  $\frac{\text{d}x}{\text{d}y}=0$ ; (ii)  $\frac{d^2y}{dx^2}\ >\ 0$  

The point at which the function has neither maximum or minimum value is called