If the curve $y = a x^{3} + b x^{2} + c x$ is inclined at $45^{0}$ to the x-axis at $(0, 0)$ but it touches the x-axis at $(1, 0)$ then the value of $a, b$ and $c$ are given by

a = - 1, b = 2, c = 1

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a = 1, b = - 2, c = 1

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a = 1, b = 1, c = - 2

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a = - 2, b = 1, c = 1

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Solution

The correct option is B $a = 1, b = - 2, c = 1$
$\frac{d y}{d x} = 3 a x^{2} + 2 b x + c$ (given)
At $(0, 0), \frac{d y}{d x} = c = tan 45^{0} = 1 \Rightarrow c = 1$
At $(1, 0), \frac{d y}{d x} = 3 a + 2 b + c$
Tangent at $(1, 0)$ is $y - 0 = (3 a + 2 b + c) . (x - 1)$
This is x-axis if $3 a + 2 b + c = 0$
which is true if $a = 1, b = - 2$
Hence $a = 1, b = - 2, c = 1$

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If the curve y=ax3+bx2+cx is inclined at 450 to the x-axis at (0,0) but it touches the x-axis at (1,0) then the value of a,b and c are given by

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