Correct option is (a) ^x^4⁄4 – 10x^3 + 150x^2 – 1000x + 10131729
The explanation: Given that f(a) = f(b) for all integral values of a, b, one additional constraint is that \(\frac{a+b}{2}\) = 10
This means that the graph is symmetric about the line x = 10
This means that the even degree polynomial has a Rolle point(the only Rolle point) at x = 10
We need to differentiate the functions given in the options and observe which of these has a single(OR repeated root), which is equal to 10.
The first Option when differentiated yields
f'(x) = x^3 – 30x^2 + 300x – 1000
Equating to zero, we see that x = 10
satisfies the equation
(10)^3 – 30(10)^2 + 300(10) – 1000 = 0.