The value of $a$ that maximizes $g(a)?$

Question

Let $X$ be a normal random variable with mean $2$ and variance $4$ and $g(a)=P(a \leq X \leq a+2)$. The value of $a$ that maximizes $g(a)?$

My thought: From the given information about random variable we get a standard normal. But after that how do I proceed?

Answer 1

Hint: We have that

$$\mathbb P(a \leq X \leq a+2)=\int_a^{a+2}\,\frac{1}{\sqrt{4\pi}}\,e^{-(x-2)^2/8}\,dx$$