Find f(x) using this information

Question

f”(x)=12x^2 - 6x + 2 and f(0)=2, f(2)=0

Answer 1

f''(x) = 12x² -6x+2

Integrate both sides to get df/dx=f '(x)

f'(x) = (1/3)12x³-(1/2)6x²+2x+c₁ = 4x³-3x²+2x+c₁

c₁ is our first constant of integration

Integrate both sides again to get f

f(x) = (1/4)4x⁴-(1/3)3x³+(1/2)2x²+c₁x+c₂=x⁴-x³+x²+c₁x+c₂

c₂ is our second constant of integration

to find our constants, plug in f(0)=2 and f(2)=0 into our f(x) equation

f(0)=2=0-0+0+0+c₂

--> c₂ = 2

f(2)=0=2⁴-2³+2²+c₁2+2=14+2*c₁

--> c₁=-7