Let x>0, y>0, x≥3-y. Find the minimum value of the expression: A= 2x²+y²+28/x + 1/y +2023

[mellow quiver]

my answer is 4³√196 +2025 and my teacher said it is incorrect

[mellow quiver]

so what is the correct answer

[shy goblet]

Did you use Lagrange multipliers?

Since the gradient has both components positive in the region, the minimum will be on the boundary

Check for x=0 and y=0

And then check for x+y=3

[spiral crescent]

How can you have x=0

[mellow quiver]

I used AM-GM

[shy goblet]

Damn right

Check x+y=3

[red horizon]

is it 1/y+2023 or 1/(y+2023) ?

[mellow quiver]

1/y +2023

[red horizon]

Just check that A=(2+7/x)(x-2)²+(1+1/y)(y-1)²+2044+x+y. So,
A≥2044+3=2047. This value is attained at (x,y)=(2,1).

[mellow quiver]

can you show me the step

[red horizon]

Which one?

[mellow quiver]

how 2x²+y²+28/x+1/y+2023 = (2+7/x)(x-2)²+(1+1/y)(y-1)²+2044+x+y

[red horizon]

just expand brackets

or copy paste it into wolfram alpha

[sonic cairn]

so minimum value is ||2047|| ?