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For the function f(x, y) = − x2 + xy + y2 + At, what value of A gives a local maximum at (-2,-4)?
- 5
- 10
- -10
- not a possible value of A
1 Answer
- 7 years ago
f(x, y) = -x^2 + xy + y^2 + At
I think "At" is a typo. Did you mean "Ax" or "Ay"?
If you meant Ax, then
fx = -2x + y + A
fy = x + 2y
fxx = -2
fxy = 1
fyy = 2
Note that fxx*fyy - (fxy)^2 < 0 for local maximum and so (-2)(2) - (1)^2 = -5 < 0.
Solve fx = fy when (x, y) = (-2, -4).
-2x + y + A = x + 2y
-2(-2) + (-4) + A = (-2) + 2(-4)
4 - 4 + A = -2 - 8
A = -10
If you meant Ay, then
fx = -2x + y
fy = x + 2y + A
fxx = -2
fxy = 1
fyy = 2
Again fxx*fyy - (fxy)^2 < 0 so we are good.
fx = fy
-2x + y = x + 2y + A
-2(-2) + (-4) = (-2) + 2(-4) + A
4 + -4 = -2 + (-8) + A
0 = -10 + A
A = 10
