Which claim is correct regarding the extrema of a function?

Question

I have a function $f$ and I know that the function has only four unique extrema values which values are fA, fB, fC, and fD; moreover, I know that

• If condition A holds then fA is a local maximum of the function
• If condition B holds then fB is a local minimum of the function
• If condition C holds then fC is a local maximum of the function
• If condition D holds then fD is a local minimum of the function

Now, I want to check for instance if fA can be a global maximum while fD is a global minimum of the function simultaneously.

Then, which claim is correct?

1. It is enough to only check if fA>fD.

2. I should check if the three conditions fA>fD, A and D can be fulfilled simultaneously.

Many thanks in advance for any help and comments.