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) $\frac{3}{4}\leq\mathrm{A}\leq 1$

We are to find the maximum and the minimum values of the function
$$f ( x ) = 4 \sin ^ { 3 } x + 4 \cos ^ { 3 } x - 8 \sin 2 x - 7$$
where $0 \leqq x \leqq \pi$.
Set $t = \sin x + \cos x$. Since
$$\sin x + \cos x = \sqrt { \mathbf { A } } \sin \left( x + \frac { \mathbf { B } } { \mathbf { C } } \pi \right) , \quad ( \text { note: have } \mathbf { B } < \mathbf { C } )$$
the range of values which $t$ takes is $- \mathbf { D } \leqq t \leqq \sqrt { \mathbf{E} }$. Next, since
$$\sin 2 x = t ^ { 2 } - \mathbf { F }$$
and
$$4 \sin ^ { 3 } x + 4 \cos ^ { 3 } x = - \mathbf { G } t ^ { 3 } + \mathbf { H } t ,$$
we have
$$f ( x ) = - \mathbf { G } t ^ { 3 } - \mathbf { I } t ^ { 2 } + \mathbf { H } t + \mathbf { J } . \tag{1}$$
When we set the right side of (1) as $g ( t )$ and differentiate with respect to $t$, we have
$$g ^ { \prime } ( t ) = - \mathbf { K } ( \mathbf { L } t - \mathbf { M } ) \left( t + \mathbf { N } \right) .$$
Hence at $t = \frac { \mathbf { O } } { \mathbf { P } } , g ( t ) ( = f ( x ) )$ takes the maximum value $\frac { \mathbf { Q R } } { \mathbf{S} }$, and at $t = \sqrt { \mathbf { U } }$, it takes the minimum value $\mathbf { V } \sqrt { \mathbf { W } } - \mathbf { X Y }$.

This question is about pairs of functions f and g that satisfy
$$\begin{aligned} f ( x ) - g ( x ) & = 2 \sin x \\ f ( x ) g ( x ) & = \cos ^ { 2 } x \end{aligned}$$
for all real numbers $x$.
Across all solutions for $\mathrm { f } ( x )$, what is the minimum value that $\mathrm { f } ( x )$ attains for any $x$ ?