Application of Derivatives

Question

The curve y = ax³ + bx² + cx + 5 touches the x -axis at P (– 2, 0) and cuts the y-axis at a point Q where its gradient is 3. Find a. b, c.

Solution

The equation of curve is
 y = ax³ + bx² + cx + 5 ...(1)
It meets y-axis at Q where x = 0
$therefore$ putting x = 0 in (1), we get,
$straight y space equals space 0 plus 0 plus 0 plus 5 space equals space 5$
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$therefore space space space space 12 straight a minus 4 straight b plus 3 space equals space 0$ ...(2)
Also (-2, 0) lies on (1)
$therefore space space 0 space equals space minus 8 straight a plus 4 straight b minus 2 straight c plus 5 space space space space space space space space space space space space space$
or $8 straight a minus 4 straight b plus 1 space equals space 0$
Subtracting (3) from (2), 4 a +2 = 0 $rightwards double arrow space space space space straight a space equals space minus 1 half$
From (2), -6 - 4b + 3 = 0 $rightwards double arrow space space space 4 straight b space equals space minus 3 space space space rightwards double arrow space space space straight b space equals space minus 3 over 4$
$therefore space space space straight a space equals space minus 1 half comma space space space space straight b space equals space minus 3 over 4 comma space space space straight c space equals 3$

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