本题目来源于试卷: Differential Calculus，类别为 IB数学
[问答题]
In a triangle $\mathrm{ABC}$, $\mathrm{BÂC}=60^{\circ}$, $\mathrm{AB}=(1-x) \mathrm{cm}$, $\mathrm{AC}=(x+3)^{2} \mathrm{~cm}$,$-3\lt x \lt 1$
1. Show that the area, $A \mathrm{~cm}^{2}$ , of the triangle is given by
$A=\frac{\sqrt{3}}{4}\left(9-3 x-5 x^{2}-x^{3}\right)$ .
2. a. Calculate $\frac{\mathrm{d} A}{\mathrm{~d} x}$ .
b. Verify that $\frac{\mathrm{d} A}{\mathrm{~d} x}=0 $ when $x=-\frac{1}{3}$ .
c. 1. Find $\frac{\mathrm{d}^{2} A}{\mathrm{~d} x^{2}}$ and hence verify that $ x=-\frac{1}{3}$ gives the maximum area of triangle A B C .
2. Calculate the maximum area of triangle A B C .
3. Find the length of [BC] when the area of triangle A B C is a maximum.
[/BC]
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