Question

a curve is such that d²y/dx²=12x-12. find the equation at point (2,9)?

Answer 1

$$\dfrac{dy}{dx}=\int(12x-12)dx=6x^2-12x+C_1$$

$$y=\int(6x^2-12x+C_1)dx=2x^3-6x^2+C_1 x+C_2$$

Point $(2,9)$

$$9=2(2)^3-6(2)^2+2C_1+C_2$$

$$C_2=17-2C_1$$

The equation of the curve is

$$y=2x^3-6x^2+C x+17-2C$$