一条方程为 $y=\mathrm{f}(x)$ 的曲线满足 $\mathrm{f}'(x)=2x^{-\frac13}-x^{\frac13}$。已知 $\mathrm{f}(8)=5$。

求 $\mathrm{f}(x)$。

Answer all questions. You should use a calculator where appropriate.

You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator.

Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.

请回答所有题目。适当时应使用计算器。必须清楚展示所有必要的计算过程；仅给出计算器答案而没有过程，不予评分。除非题目另有精度要求，非精确数值答案保留 3 位有效数字，以度为单位的角度保留 1 位小数。

逐项积分后用 $\mathrm f(8)=5$ 求常数。计算分数幂时先取立方根。

On the interval containing $8$, integration gives

At $x=8$, the first two terms give $3(4)-\tfrac34(16)=0$, so $C=5$. Hence $\mathrm f(x)=3x^{2/3}-\tfrac34x^{4/3}+5$ on $x>0$.