(i)

(a) 求 $f'(x)$，并写成 $\dfrac{P(x)}{Q(x)}$ 的形式，其中 $P(x)$ 和 $Q(x)$ 都是完全因式分解后的二次式。

(b) 由此求 $f(x)$ 为增函数时 $x$ 的取值范围。

(ii)

曲线 $y=g(x)$ 在点 $M$ 处取得最大值。

证明点 $M$ 的 $x$ 坐标满足方程

其中 $k$ 为待求常数。

用商法则求导，再从分子提取公因式。

We have

Use the quotient rule:

Now

and

So

Therefore

先将分式拆成多项式与一个简单分式，逐项求导后再通分。

Since $2x+5=2(x-3)+11$, expanding the square and dividing by $x-3$ gives

Differentiating and putting the result over a common denominator,

对乘积和根式分别用乘积法则与链式法则。极大值点在区间内部，且该点的余弦因子非零。

Let $x_M$ be the $x$-coordinate of $M$. Since $g(0)=0$ and $g(x)>0$ for $0<x<\pi/4$, the maximum occurs at an interior point.

Using the product rule and chain rule,

At $M$, set $g'(x_M)=0$ and multiply by $\sqrt{\sin 4x_M}$:

If $\cos 4x_M=0$, then $x_M=\pi/8$ and the left-hand side is $1$, so this is impossible. Dividing by $\cos 4x_M$ gives

so $\boxed{k=2}$.

函数在给定区间非负，最大值的位置在平方后不变。先平方，可以避免对根式直接求导。

On the given interval $g(x)\geq0$. Its maximum occurs at the same point as the maximum of

The maximum is interior because $H(0)=0$ and $H(x)>0$ for $0<x<\pi/4$. At its $x$-coordinate $x_M$,

As $x_M>0$, divide by $2x_M$ to get $\sin 4x_M+2x_M\cos 4x_M=0$. The cosine factor cannot be zero, since then the sine would be $1$. Therefore