Incredibox - Sprunki

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Let f(x)f(x) be a twice-differentiable function such that

∫1x(t3−2t+5)f′′(t) dt=x5−7x3+4x+C\int_1^x (t^3 - 2t + 5) f''(t) \,dt = x^5 - 7x^3 + 4x + C

where C is a constant.

1. Determine f′′(x)f''(x).
2. Given that f′(x)=ax3+bx2+cx+df'(x) = ax^3 + bx^2 + cx + d, find the constants a,b,c,da, b, c, d.
3. Suppose f(x)f(x) satisfies the functional equation

f(Ax+B)=Af(x)+Bf(Ax + B) = A f(x) + B

for some real numbers AA and BB. Find all possible values of AA and BB such that f(x)f(x) is a polynomial function.

1. Solve for all real roots of f(x)f(x) given that f(1)=2f(1) = 2 and f(2)=5f(2) = 5.2

do it. 

now.