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Chapter 12: Q. 6 (page 975)

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

Short Answer

Expert verified

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 by step solution

01

Step 1. Given 

Saddle point and critical point .

02

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 (x0,y0)鈥�at which 鈭�f(x0,y0)鈥�=0and the discriminate of the function is negative that is Hf(x0,y0)鈥�<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.

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