题目

Problem

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

\begin{align*} y=x^3+96\sqrt{x}+5,\qquad x>0 \end{align*}

(a) Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ , giving each term in simplest form.

(3)

(b) Find the solution of the equation

\begin{align*} \frac{\mathrm{d}^2y}{\mathrm{d}x^2}=0 \end{align*}

writing the answer in the form $2^k$ where $k$ is a constant.

(3)

题目中文翻译

本题必须写出解题过程的所有步骤。

不得使用依赖计算器技术的解法。

\begin{align*} y=x^3+96\sqrt{x}+5,\qquad x>0. \end{align*}

(a) 求 $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ ，并将每一项写成最简形式。

(3)

(b) 解方程

\begin{align*} \frac{\mathrm{d}^2y}{\mathrm{d}x^2}=0, \end{align*}

并将答案写成 $2^k$ 的形式，其中 $k$ 为常数。

(3)

解答

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把 $\sqrt{x}$ 写成 $x^{1/2}$ ，再逐项求导。最后把 $48x^{-1/2}$ 写回分式根式形式。

展开

\begin{align*} y=x^3+96x^{1/2}+5. \end{align*}

Differentiate term by term:

\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} =&\,3x^2+96\cdot\frac12x^{-1/2}+0\\ =&\,3x^2+48x^{-1/2}\\ =&\,3x^2+\frac{48}{\sqrt{x}}. \end{align*}

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先对 (a) 的导数再求一次导。令二阶导数等于 $0$ 后，整理成 $x^{5/2}=4$ ，再把 $4$ 写成 $2^2$ ，用指数法则得到 $x=2^{4/5}$ 。

展开

From part (a),

\begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x}=3x^2+48x^{-1/2}. \end{align*}

Differentiate again:

\begin{align*} \frac{\mathrm{d}^2y}{\mathrm{d}x^2} =&\,6x+48\left(-\frac12\right)x^{-3/2}\\ =&\,6x-24x^{-3/2}\\ =&\,6x-\frac{24}{x^{3/2}}. \end{align*}

Set this equal to zero:

\begin{align*} 6x-\frac{24}{x^{3/2}}=&\,0\\ 6x=&\,\frac{24}{x^{3/2}}\\ x=&\,\frac{4}{x^{3/2}}. \end{align*}

Since $x>0$ , multiply by $x^{3/2}$ :

\begin{align*} x^{5/2}=&\,4\\ x=&\,4^{2/5}\\ =&\,(2^2)^{2/5}\\ =&\,2^{4/5}. \end{align*}

Therefore

\begin{align*} x=2^{4/5}. \end{align*}