Question

Find a curve y=f(x) with the following properties (i) y"=6x and (ii) its graph passes through the point (1,1) and has a horizontal tangent there.

Answer 1

First solve the function by integrating the 2nd derivative of the function two times. Since the y’’ = 6x

Integrating

Y’ =∫ 6x

y’ = 6 (x^2)/2

y’ = 3(x^2) + c

integrating it further to solve the function

y =∫ 3(x^2) + c

y = 3(x^3)/3 + c1

y = (x^3) + c1

to solve c1

substitute the point (1,1)

1 = (1^3) + c1

C1 = 0

So the function is

y = x^3