The condition that $$f(x)=a x^3+b x^2+c x+d$$ has no extreme value is

At any point $$(x, y)$$ on a curve if the length of the subnormal is $$(x-1)$$ and the curve passes through $$(1,2)$$, then the curve is a conic. A vertex of the curve is

If $$y=4 x-6$$ is a tangent to the curve $$y^2=a x^4+b$$ at $$(3,6)$$, then the values of $$a$$ and $$b$$ are

Find the positive value of $$a$$ for which the equality $$2 \alpha+\beta=8$$ holds, where $$\alpha$$ and $$\beta$$ are the points of maximum and minimum, respectively, of the function $$f(x)=2 x^3-9 a x^2+12 a^2 x+1$$.