Question

Let f(x,y) = x^2 + y^2 + 8xy + 75y + 10. Find all the critical points and determine if each is a maxima, minima, or saddle points.

Answer 1

Solution: Given a function:

The objective is to find and classify the critical points of the given function.

Partially differentiating the function w.r.t x and y:

... (1)

and

... (1)

Solving the linear equations (1) and (2), we'll get the critical point:

Now, we'll use the second derivative test which states that:

• if at any critical point,

Then the function will have a saddle point here.

• If at any critical point,

Also, if,

Then the function will have a minimum here.

Else if,

Then the function will have a minimum here.

Further differentiating equations (1) and (2), we'll get:

Also, partially differential any of the equation (1), w.r.t y, or (2), w.r.t x, to calculate the mixed order partial derivative, we'll get,

Therefore,

Therefore, the critical point calculated is a saddle point for the given function.