Relative Minimums and Maximums — Question 1

Question 1 – Solution

Strategy. Solve the first-derivative equations, classify with the Hessian determinant, and then use an exact algebraic form as an independent check.

Step 1: Critical point $$f_x=2x-4,\qquad f_y=4y+8.$$ Both vanish only when $$\boxed{(x,y)=(2,-2)}.$$

Step 2: Second derivative test $$f_{xx}=2,\qquad f_{yy}=4,\qquad f_{xy}=0.$$ The Hessian determinant is $$D=f_{xx}f_{yy}-(f_{xy})^2=8>0.$$ Since $f_{xx}>0$, the point is a strict relative minimum. Its value is $$f(2,-2)=\boxed{-11}.$$

Step 3: Algebraic verification Completing squares gives $$f(x,y)=(x-2)^2+2(y+2)^2-11.$$ Both squared terms are nonnegative and vanish simultaneously only at $(2,-2)$. This directly confirms the strict local minimum and rules out any other critical point or relative extremum.