题目

Problem

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

Solutions relying on calculator technology are not acceptable.

A curve has equation

\begin{align*} y=\frac{4x^2+9}{2\sqrt{x}},\qquad x>0. \end{align*}

Find the $x$ coordinate of the point on the curve at which $\dfrac{dy}{dx}=0$ .

(6)

解答

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先把分式拆成两项，并写成指数形式，再求导。因为 $x>0$ ，最后只保留正的 $x$ 值。

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\begin{align*} y =&\,\frac{4x^2}{2x^{1/2}}+\frac{9}{2x^{1/2}}\\ =&\,2x^{3/2}+\frac92x^{-1/2}. \end{align*}

Differentiate:

\begin{align*} \frac{dy}{dx} =&\,3x^{1/2}-\frac94x^{-3/2}. \end{align*}

Set $\dfrac{dy}{dx}=0$ :

\begin{align*} 3x^{1/2}-\frac94x^{-3/2}=&\,0\\ 3x^{1/2}=&\,\frac94x^{-3/2}. \end{align*}

Multiply by $x^{3/2}$ :

\begin{align*} 3x^2=&\,\frac94\\ x^2=&\,\frac34. \end{align*}

Since $x>0$ ,

\begin{align*} x=\frac{\sqrt3}{2}. \end{align*}