题目

Problem

Solutions based entirely on graphical or numerical methods are not acceptable.

Given

\begin{align*} f(x)=2x^{5/2}-40x+8,\qquad x>0, \end{align*}

(a) solve the equation $f'(x)=0$ .

(4)

(b) solve the equation $f''(x)=5$ .

(3)

解答

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先求一阶导数，再令它等于 $0$ 。由于 $x>0$ ，只保留正解。

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\begin{align*} f'(x)=&\,5x^{3/2}-40. \end{align*}

Set $f'(x)=0$ :

\begin{align*} 5x^{3/2}-40=&\,0\\ x^{3/2}=&\,8. \end{align*}

Since

\begin{align*} 4^{3/2}=8, \end{align*}

the solution is

\begin{align*} x=4. \end{align*}

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由 $f'(x)$ 再求一次导，令二阶导数等于 $5$ 。

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\begin{align*} f''(x)=&\,\frac{15}{2}x^{1/2}. \end{align*}

Set $f''(x)=5$ :

\begin{align*} \frac{15}{2}x^{1/2}=&\,5\\ x^{1/2}=&\,\frac23\\ x=&\,\frac49. \end{align*}