A Cosine Difference Maximum

Let $f(x)=5\cos x-\cos(5x)$.

1. Find the maximum value of $f(x)$ on $\left[0,\dfrac{\pi}4\right]$. 2. Given $\theta\in(0,\pi)$ and a real number $a$, prove that there exists $y\in[a-\theta,a+\theta]$ with $\cos y\leqslant\cos\theta$. 3. If there exists $\varphi\in\mathbb R$ such that $5\cos x-\cos(5x+\varphi)\leqslant b$ for all $x$, find the minimum value of $b$.