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

(17 points)
(1) Find the maximum value of the function $f(x) = 5\cos x - \cos 5x$ on the interval $\left[0, \frac{\pi}{4}\right]$.
(2) Given $\theta \in (0, \pi)$ and $a \in \mathbf{R}$, prove that there exists $y \in [a - \theta, a + \theta]$ such that $\cos y \leq \cos \theta$.
(3) Let $b \in \mathbf{R}$. If there exists $\varphi \in \mathbf{R}$ such that $5\cos x - \cos(5x + \varphi) \leq b$ holds for all $x \in \mathbf{R}$, find the minimum value of $b$.

Let $\alpha$, $\beta$ and $\gamma$ be the angles of an acute angled triangle. Then the quantity $\tan\alpha\tan\beta\tan\gamma$
(a) Can have any real value
(b) Is $\leq 3\sqrt{3}$
(c) Is $\geq 3\sqrt{3}$
(d) None of the above.

If the maximum and minimum values of $\sin ^ { 6 } x + \cos ^ { 6 } x$, as $x$ takes all real values, are $a$ and $b$, respectively, then $a - b$ equals
(A) $\frac { 1 } { 2 }$.
(B) $\frac { 2 } { 3 }$.
(C) $\frac { 3 } { 4 }$.
(D) 1 .

If $A=\sin^{2}x+\cos^{4}x$, then for all real $x$
(1) $\frac{13}{16}\leq\mathrm{A}\leq 1$
(2) $1\leq A\leq 2$
(3) $\frac{3}{4}\leq\mathrm{A}\leq\frac{13}{16}$
(4) $\frac{3}{4}\leq\mathrm{A}\leq 1$