Q. 6 - Calculus

Question

Explain why every saddle point is both a stationary point and a critical point. 

Step-by-Step Solution

Answer

 A saddle point is both a stationary point as well as a critical point as the function is differentiable at the saddle point and the gradient vanishes as well. 

Step 1. Given

Saddle point and critical point .

Step 2. Explanation .

For a function of two variable f(x ,y) the saddle point is a stationary point in the domain of  f(x ,y)  at which the function neither have a maximum nor a minimum. That is a saddle point is a point  $(x_0 , y_0)\hspace{0.17em}$at which $\nabla f(x_0 , y_0)\hspace{0.17em}=0$ and the discriminate of the function is negative that is  $H_{f }(x_0 , y_0)\hspace{0.17em}<0$ .

So a saddle point satisfies both the conditions for a point to be a critical point. Hence a saddle point is both a stationary point as well as a critical point as the function is differentiable at the saddle point and the gradient vanishes as well.