First slide

Tangents and normals

Question

$The curve y = a x^{3} + b x^{2} + c x + 8 touches x -axis at P (- 2, 0) and cuts the$ $y-axis at a point Q where its gradient is 3. The values of a, b, c are respectively$

Moderate

A

$- 1 / 2, - 3 / 4, 3$
B

$3, - 1 / 2, - 4$
C

$- 1 / 2, - 7 / 4, 2$
D

none of these

Solution

$\begin{array}{l} \frac{d y}{d x} = 3 a x^{2} + 2 b x + c \\ Since the curve touches x -axis at (- 2, 0) so \end{array}$
${\frac{d y}{d x}|}_{(- 2, 0)} = 0 \Rightarrow 12 a - 4 b + c = 0 - - - - (i)$
The curve cut the y-axis at (0, 8) so
${\frac{d y}{d x}|}_{(0, 8)} = 3 \Rightarrow c = 3$
$\begin{array}{l} Also the curve passes through (- 2, 0) so \\ 0 = - 8 a + 4 b - 2 c + 8 \Rightarrow - 8 a + 4 b - 2 = 0 - - - - (i i) \end{array}$
$\begin{aligned} Solving (i) and \\ (ii) a = - 1 / 4, b = 0 \end{aligned}$

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