Question

The function f(x, y) = 10−x 2−4y 2+2x has one critical point. Find that critical point and show that it is not a saddle point. Indicate whether this critical point is a maximum or a minimum, and find that maximum or minimum value.

Answer 1

Solution: The given function is:

To find the critical point of the given function, we will first partially differentiate the given function with respect to and and then equate the results obtained equal to zero, i.e., and . Calculating the first partial derivative with respect to , we get:

Equating the above equal to zero, we get:

Similary,

Equating the above equal to zero, we get:

Therefore, the critical point for the given function is at . In order to check whether the critical point is a saddle point, minimum point or a maximum point, we need to perform the Second derivative test. For this, we need to find out , and  . Now, we can write determinant of Hessian matrix using the obtained second partial derivatives as:

The conditions of the Second derivative test are:

(i) If , or  , then the critical point is the minimum point.

(ii) If ,   or  , then the critical point is the maximum point.

(ii) If , then the critical point is the saddle point.

Now we will find the required second derivatives. Partially differentiating equation with respect to , we get:

Partially differentiating equation with respect to , we get:

Partially differentiating the given function with respect to and ,we get:

Substituting the values from , and in the Hessian matrix, we get:

Since we get and , therefore, the point is not a saddle point. In fact it is the point of maximum.

The value of maximum can be found by substituting in the given function . Therefore, we get: