Exponential function, multivariable calculus

Question

I got the function $f(x,y)=e^{-x^2-y^2}$ with the domain of definition $x^2+y^2\leq25$. The task is to decide the biggest and lowest value. How do I get there?

Answer 1

$F(x,y)=e^{-x^2-y^2}=e^{-(x^2+y^2)}$. Note that $e^{-z}$ is strictly decreasing with respect to $z$. So to maximize and minimize $F(x,y)$, just minimise and maximise $x^2+y^2$ respectively. By the domain of definition, $x^2+y^2$ is minimised at $0$ and maximised at $25$, so the maximum and minimum values of $F(x,y)$ are $e^{-0}=1$ and $e^{-25}$, respectively.

Answer 2

The minimum or maximum must lie either in the interior region, or on the boundary.

For the interior, you find where the derivative of the function is zero, for both $x$ and $y$.

$\frac{d}{dx}e^{-x^2-y^2} = -2xe^{-x^2-y^2} = 0 $ =>

$x=0$ and similarily $y=0$.

This means we have a possible min or max in $f(0,0) = 1$

Now for the boundary. You can realise the function is constant on all circles with center in (0,0) by inspecting the function closely. $f(x,y)=e^{-x^2-y^2}=e^{-(x^2+y^2)}$, and $x^2+y^2$ is constant on a circle. That means you only have to evaluate one boundary point.

I have chosen $(0,5)$ as my boundary point, and the value of f is $f(0,5)=e^{-25}$.

The minimum and maximum are thus $e^{-25}$ and 1.

An alternative way of solving the boundary would be to parameterise the boundary, and finding the min and max of this function.

Hope this explains it for you.