A curve has equation$$y=x^3+ax^2+bx+c.$$ At $x=1$, its tangent is $$y=2x+1.$$ Also, \[\left.\frac{

Question

A curve has equation$$y=x^3+ax^2+bx+c.$$ At $x=1$, its tangent is $$y=2x+1.$$ Also, $$\left.\frac{\mathrm{d}^2y}{\mathrm{d}x^2}\right|_{x=1}=0.$$ Find $a,b,c$. Show that the curve has no horizontal tangent anywhere.

Accepted Answer

(a) $a=-3,\ b=5,\ c=0$; (b) $y'=3(x-1)^2+2>0$.

Differentiation

Differentiation