A curve is such that $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=kx+2$, where $k$ is a const

Question

A curve is such that $\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}=kx+2$, where $k$ is a constant. The gradient is $-1$ when $x=0$ and $7$ when $x=2$. Given that the curve passes through $(1,3)$, find the value of $k$ and the equation of the curve.[8]

Accepted Answer

$k=2,\quad y=\dfrac{x^3}{3}+x^2-x+\dfrac83$

Parameterised second derivative and gradient data

Parameterised second derivative and gradient data

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