To find the first derivative of f(x), we need to integrate the second derivative with respect to x once.
f'(x) = ∫ f''(x) dx = ∫ x(x-a)(x-b)^2 dx
f'(x) = ∫ (x^4 - (a+b)x^3 + (a^2+3ab+b^2)x^2 - ab(a+b)x + ab^2) dx
f'(x) = 1/5 x^5 - 1/4(a+b)x^4 + 1/3(a^2+3ab+b^2)x^3 - 1/2ab(a+b)x^2 + 1/3ab^2x + C
where C is the constant of integration.
To find the second derivative of f(x), we need to differentiate f'(x) with respect to x.
f''(x) = d/dx [1/5 x^5 - 1/4(a+b)x^4 + 1/3(a^2+3ab+b^2)x^3 - 1/2ab(a+b)x^2 + 1/3ab^2x + C]
f''(x) = 1 x^4 - 1(a+b)x^3 + 1(a^2+3ab+b^2)x^2 - 1ab(a+b)x + 1ab^2
f''(x) = x^4 - (a+b)x^3 + (a^2+3ab+b^2)x^2 - ab(a+b)x + ab^2
Therefore, the second derivative of f is given by f''(x) = x^4 - (a+b)x^3 + (a^2+3ab+b^2)x^2 - ab(a+b)x + ab^2.