I'm having trouble finding the local minimum and maximum of the next functions:
$$1. f(x,y) = e^{x+y^2} \cdot y $$
$ f_x'= (e^{x+y^2}\cdot y) ; $ $ f_y'= (e^{x+y^2}(1+2y^2)) $
$$ 2. f(x,y) = e^{x^2-y^2}\cdot y $$
$ f_x'= (2xy \cdot e^{x^2+y^2}) ; $ $ f_y'= (e^{x^2+y^2}-y^3 \cdot e^{x^2+y^2}) $
I have found the derivative of $f_x$ and $f_y$. But since I cant find any suspicious min and max for $x$ and $y$ i'm stuck. How i can extract $x$ and $y$ values?
If i had my suspicious points I would put them in Hessian matrix and find the points.
Any help would really appreciated!
Best Answer
To find extrema, you search for where both the partial derivatives are zero.
As you (almost) wrote, $f_x=e^{x+y^2}\cdot y$. This is zero only if $y=0$. However, $f_y=e^{x+y^2}\cdot (2y^2+1)$ is then non-zero. Therefore there is no critical point for this function, and it has no local minima or maxima on the entire $xy$-plane.
Now do the same kind of analysis on the second function.
Note that you wrote "I can't find any suspicious min and max for $x$ and $y$." But you are not looking for min of $x$ or $y$: you are looking $x$ and $y$ where both partial derivatives are zero. Sometimes such a place does not exist, as in this problem. You should check such a problem with a three-dimensional graphing program or calculator. WolframAlpha is one place for such graphing.