Continuous game payoff

Question

I'm new in game theory. The lecturer said that is something wrong with a sentence that is highlighted in yellow, but I can't spot any inaccuracy in it. Please share your opinion, what is wrong in that sentence.

Answer 1

The fragment requires only continuity of $p_2$, but then differentiates it - which would require $p_2$ to be differentiable, which is stronger property. Also, it assumes that function achieves maximum in only one point, which in general requires function to be monotonic.

Answer 2

Even if it is differentiable, the maximum might be at the endpoint value of $b$