A Sine-Composed Quadratic

Let $f(x)=x^2-ax+b$.

1. Discuss the monotonicity of $f(\sin x)$ on $\left(-\dfrac{\pi}2,\dfrac{\pi}2\right)$, and determine whether it has extreme values (finding them when they exist). 2. Let $f_0(x)=x^2-a_0x+b_0$. Find the maximum value $D$ of $\left|f(\sin x)-f_0(\sin x)\right|$ on $\left[-\dfrac{\pi}2,\dfrac{\pi}2\right]$. 3. In part 2, take $a_0=b_0=0$. Find the maximum value of $z=b-\dfrac{a^2}4$ subject to $D\leqslant 1$.