Find f. f ''(x) = x−2, x > 0, f(1) = 0, f(6) = 0

Question

Find f. f ''(x) = x−2, x > 0, f(1) = 0, f(6) = 0

Answer 1

To find f(x), two integration steps (indefinite integrals) are needed. That's why two points are given for f(x).

Integrate once:

Integrate again to find f(x):

Now you need to find c₁ and c₂, but you know f(1)=0 and f(6)=0:

Solving two equations in two unknowns (for c₁ and c₂), you find c₁=-1/6 and c₂=1

Final answer:

You can check that f(1)=0, f(6)=0 and that the second derivative of this function is equal to x-2.