I have to use the intermediate value theorem (IVT) to show that the equation $\cos(x+y) + \sin(xy) = x^2 + y$
- Has infinitely many solutions in $x,y$, and
- The set of all solutions is a closed set in $\mathbb{R}^2$
So far, I have plotted the equation (link), and read similar posts here and here but I am not sure how to proceed. Any help would be greatly appreciated!
PS: One hint I got was to fix $y$ and apply the IVT repeatedly. So I fixed $y=0.5$ and plotted $f(x) = \cos(x+0.5) + \sin(0.5x) - x^2 - 0.5$ (link). I noticed that when I fix $y$ such that $y<1$, the function intersects (or is tangent too) the $x$ axis, whereas for $y \geq 1$, it does not touch the axis. I'm not sure how to use this to prove 1 and 2 above.