# Need help finding examples of pairs of functions satisfying certain conditions

I want to find pairs of two functions on two variables $$f_1(x,y)$$ and $$f_2(x,y)$$, such that each one can be minimized on one variable ($$f_1$$ needs to have a minimum on the variable $$x$$ and $$f_2$$ needs to have a minimum on $$y$$) but in such a way that their gradients are bounded, and also their graphs are bounded, and such that the minima of each happen in distinct points $$(\bar{x},\bar{y})$$.

For instance, if I take $$f_1(x,y) = (x-y)^2$$, the minimum happens when $$x=y$$, and if I take $$f_2(x,y) = (x-(y-1))^2$$, the minimum happens when $$x=y-1$$. The minima don't have intersection, but the functions and gradient are not bounded. I tried doing something involving sine and cosine but the problem is that there too many minima, I'd rather them have only one, or at least a global minimum...

Are there any examples satisfying all the conditions?

• What does "limited" mean in this context? Did you mean to say "bounded"? If so, maybe a function like $\exp(-x^2)$ does what you want, as it has a unique critical point and the derivative is obviously bounded. of course, you'll have to modify it slightly to get the sort of functions you specified.
– lulu
Oct 13 at 23:45
• Ah, yes. I meant bounded, thx. While exp(-x^2) is bounded, it does not admit a point on whih it attains the minimum, tho... Oct 14 at 0:47
• It has a unique critical point, which is all that matters. Trivial modifications are needed to match your criteria (for instance, look at $-\exp(-x^2)$ instead).
– lulu
Oct 14 at 11:12