I have been trying to find this multiplexer's truth table but I have had problems with (B1+B0)'.(A0'.A1') as you can see from the image below.
Might someone help me out?

(Sorry for the Typo in the title, I can't seem to edit it)

 

Cropped picture for convenience:


How many inputs and outputs does the MUX (assume it's 4 to 1) have?

Try labeling the inputs and drawing the truth table first.

 

You need to indicate which input is associated with which multiplexer select setting. Don't make readers guess. A common way to do this is to put a '0' on the input that is selected when all of the select inputs are LO. The convention is that this is one of the end inputs and then the others walk up numerically if the input is treated as a multibit integer with a0 being the lsb. Your K-map labels should use these in the same order.

But your either didn't order your inputs in numerical order, or you did your K-map wrong. Your map says that the output is always a 1 when input '11' is selected. But that would be the last input which your diagram shows being connected to b1.

As for the input with the NOR gate, first figure out what the value at the MUX input is for the various values of b0 and b1 and then enter those in the table on the appropriate row.