# Maximum and Minimum of multivariable calc

• Mar 17th 2009, 08:12 PM
acg716
Maximum and Minimum of multivariable calc
Find the absolute maximum and minimum values of f on the set D.
f(x, y) = xy^2 + 9D = {(x, y) | x ≥ 0, y ≥ 0, x^2 + y^2 ≤ 3}
• Mar 17th 2009, 09:22 PM
matheagle
I think you can just use common sense here.
The 9 does NOT affect the min and the max.
I now see that $x\ge 0$ and $y\ge 0$,
so the min is 9 which occurs along the x axis and the y axis.
Next insert the constraint in the function to get the max along the circle in the first quadrant.
$f(x)=xy^2+9=x(3-x^2)+9=3x-x^3+9$.
Now maximize this wrt x and you get x=1, since we're only in the first quadrant.
That gives $y^2=2$ and a max of 11.

I also got this via LaGrange, to check my work.
• Mar 17th 2009, 09:34 PM
acg716
So the minimum is when x and y are both -sqrt(3)?
I'm a bit confused.
• Mar 17th 2009, 09:42 PM
matheagle
Quote:

Originally Posted by acg716
So the minimum is when x and y are both -sqrt(3)?
I'm a bit confused.

Nope, I was using the entire circle, my bad.
Since and and y are nonnegative, the smallest this can be is 9 when either is zero.
• Mar 17th 2009, 10:15 PM
acg716
sorry i meant maximum!