Question

Determine whether there is a relative maximum, a relative minimum, a saddle point, or insufficient information to determine the nature of the function \(f(x, y)\) at the critical point \(\left(x_{0}, y_{0}\right)\). $$ f_{x x}\left(x_{0}, y_{0}\right)=-9, \quad f_{y y}\left(x_{0}, y_{0}\right)=6, \quad f_{x y}\left(x_{0}, y_{0}\right)=10 $$

Short answer

The critical point \((x_{0}, y_{0})\) is a saddle point.

Step-by-step solution

Compute the Hessian Determinant

With \(f_{xx} = -9\), \(f_{yy} = 6\) and \(f_{xy} = 10\), let's compute the Hessian determinant using the formula \(D = f_{xx}f_{yy} - (f_{xy})^2\). Plugging in these values, we get \(D = -9*6 - (10)^2\).

Evaluate the Hessian Determinant

Calculate the value of D, \(D = -54 - 100 = -154\).

Find the nature of the critical point using the Hessian Determinant

Since D < 0, we conclude, based on the rules outlined in the analysis, that the critical point \((x_{0}, y_{0})\) is a saddle point.