Question

Find a function f such that the curve y = f(x) satisfies y'' = 12x, passes through the point (0,1), and has a horizontal tangent there.

I can use the anti-derivatives to get

y = 2x^3 + Cx + D

but I don't know how to get C even though I know D = 1.

Answer 1

you know that
y' = 6x + C
and that y'(0) = 0, so
0 = C

y = 2x^3 + 1

Answer 2

So y'(0) = 0 because there is a horizontal tangent at x = 0?