The diagram shows part of the curve with equation $$\(y=x^{3}-2 b x^{2}+b^{2} x\)$$ and the line $$\(O A\)$$, where $$\(A\)$$ is the maximum point on the curve. The $$\(x\)$$-coordinate of $$\(A\)$$ is $$\(a\)$$ and the curve has a minimum point at $$\((b, 0)\)$$, where $$\(a\)$$ and $$\(b\)$$ are positive constants. Show that $$\(b=3 a\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_s20_qp_13 Year：2020 Question No：11(a)
Answer:
$\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{2}-4 b x+b^{2}$
$\begin{array}{l}3 x^{2}-4 b x+b^{2}=0 \rightarrow(3 x-b)(x-b)(=0) \\ x=\frac{b}{3} \text { or } b\end{array}$
$a=\frac{b}{3} \rightarrow b=3 a \quad \mathbf{A G}$
Alternative method for question 11(a)
$\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{2}-4 b x+b^{2}$
Sub $b=3 a \&$ obtain $\frac{\mathrm{d} y}{\mathrm{~d} x}=0$ when $x=a$ and when $x=3 a$
$\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=6 x-12 a$
$<0$ Max at $x=a$ and $>0$ Min at $x=3 a$. Hence $b=3 a$ AG
$\begin{array}{l}3 x^{2}-4 b x+b^{2}=0 \rightarrow(3 x-b)(x-b)(=0) \\ x=\frac{b}{3} \text { or } b\end{array}$
$a=\frac{b}{3} \rightarrow b=3 a \quad \mathbf{A G}$
Alternative method for question 11(a)
$\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{2}-4 b x+b^{2}$
Sub $b=3 a \&$ obtain $\frac{\mathrm{d} y}{\mathrm{~d} x}=0$ when $x=a$ and when $x=3 a$
$\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=6 x-12 a$
$<0$ Max at $x=a$ and $>0$ Min at $x=3 a$. Hence $b=3 a$ AG
Knowledge points:
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
1.7.4 locate stationary points and determine their nature, and use information about stationary points in sketching graphs. (Including use of the second derivative for identifying maxima and minima; alternatives may be used in questions where no method is specified.) (Knowledge of points of inflexion is not included.)
Solution:
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