Question

Solve the q in the img below

Answer 1

`(-1,(1)/(2))` is a maximum

Explanation:

`\displaystyle f(x,y)=9-2x+4y-x^2-4y^2\\\\(\partial f)/(\partial x)=-2-2x\rightarrow-2-2x=0\rightarrow x=-1\\\\(\partial f)/(\partial y)=4-8y\rightarrow4-8y=0\rightarrow y=(1)/(2)`

`\displaystyle (\partial^2 f)/(\partial x^2)=-2\\\\ (\partial^2 f)/(\partial y^2)=-8\\\\ (\partial^2 f)/(\partial x\partial y)=0`

`\displaystyle H=\biggr((\partial^2 f)/(\partial x^2)\biggr)\biggr((\partial^2 f)/(\partial y^2)\biggr)-\biggr((\partial^2 f)/(\partial x\partial y)\biggr)^2=(-2)(-8)-0^2=16 > 0`

Since
`H > 0` and
`\displaystyle (\partial^2 f)/(\partial x^2) < 0`, then
`(-1,(1)/(2))` is a maximum.