Global minima problem

[Archimedess]

Homework Statement:

Let $f(x,y)=\arctan(4\sin^2(y)+3\ln(x^2+1))$ show that it has $\infty$ global minima

Relevant Equations:

No relevant equations

I'm always struggling understand how to determine if a two variable function has global minima, I know that if I find a local minima and the function is convex than the local minima is also a global minima, in this case is really difficult to determine if the function is convex.

Sorry if I don't post any attempt but I got no clue how to do this.

 

[fresh_42]

I'm afraid you will have to calculate the partial derivatives and consider where they vanish. Also look up the Hesse matrix.

 

[BvU]

no clue how to do this.

Not good enough per the PF guidelines !
Least you could do is find and discuss a few minima, remark that $\sin^2$ is periodical, etc ...

 

[RPinPA]

No, it is not convex. A convex function has a UNIQUE global minimum.

Edit: OK, I'm going to back off on that statement. That's true for a strictly convex function. But you could imagine a convex function whose set of global minima was a finite flat region. For instance, a function with $f(x,y) = 0$ on the circle of radius 1, and positive elsewhere. That would qualify as a convex function with infinitely many global minima.

But that's not the reason for infinitely many minima here. @BvU has already identified the reason in comment #3.

To find the global minima, you're going to have to identify all the local minima. Then analyze them and find which subset of those have minimal values of $f(x, y)$.