Find the extremum of f(x, y) subject to the given constraint, and state whether it is a maximum or a minimum.
f ( x, y )=2 x^2 +3y^2 - 4 xy; x+ y =27
there is a minimum value of 162 located at (x, y)=( 15,12 ).

The Lagrangian is

with critical points whenever

Solving for and , we get

so there is one critical point at (15, 12), at which point .

The Hessian matrix for this function is

with determinant 8 > 0, and , which tells us this point is a minimum.

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step-by-step explanation: