The curve $C$ has equation $$y=x^3+ax^2+bx+c,$$ where $a,b,c$ are constants. The tangent to

Question

The curve $C$ has equation $$y=x^3+ax^2+bx+c,$$ where $a,b,c$ are constants. The tangent to $C$ at the point where $x=1$ has equation $y=2x+1$. The normal to $C$ at the point where $x=-1$ is parallel to the line $x+10y=4$. Form three linear equations in $a,b,c$, and hence find their values.[5] Show that the gradient of $C$ is always positive. Hence explain why $C$ intersects the $x$-axis exactly once, and find the $x$-coordinate of this intersection correct to 3 significant figures.[3]

Accepted Answer

(i) $a=-2,\ b=3,\ c=1$ (ii) $x=-0.276$ (3 s.f.)

A cubic determined by a tangent and a normal

A cubic determined by a tangent and a normal

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