Question

Find the critical point(s) for f(x, y) = 4x² + 2y² − 8x - 8y-1. For each point determine whether it is a local maximum. a local minimum, a saddle point, or none of these. Use the methods of this class. (6 pts)

Answer 1

(1,2) is a local minimum

Explanation:

`\displaystyle f(x,y)=4x^2+2y^2-8x-8y-1\\\\(\partial f)/(\partial x)=8x-8\rightarrow 8x-8=0\rightarrow x=1\\\\(\partial f)/(\partial y)=4y-8\rightarrow 4y-8=0\rightarrow y=2\\\\\\(\partial^2 f)/(\partial x^2)=8,\,(\partial^2 f)/(\partial y^2)=4,\,(\partial^2 f)/(\partial x\partial y)=0\\\\H=\biggr((\partial^2f)/(\partial x^2)\biggr)\biggr((\partial^2 f)/(\partial y^2)\biggr)-\biggr((\partial^2 f)/(\partial x\partial y)\biggr)^2=(8)(4)-0^2=32 > 0`

Since the value of the Hessian Matrix is greater than 0, then (1,2) is either a local maximum or local minimum, which can be tested by observing the value of
`\displaystyle (\partial^2 f)/(\partial x^2)`. Since
`\displaystyle (\partial^2 f)/(\partial x^2)=8 > 0`, then (1,2) is a local minimum