If g'(3/2) = g'(1/2) and f(x) = 1/2 [g(x) + g(2-x)] and

Question

If \(g'\left(\frac {3}{2}\right) = g'\left(\frac {1}{2}\right)\) and \(f(x) = \frac {1}{2} [g (x) + g(2-x)]\) and \(f'\left(\frac {3}{2}\right) = f'\left(\frac {1}{2}\right)\) then

(1) f''(x) = 0 has exactly one root in (0, 1) 

(2) f''(x) = 0 has no root in (0, 1) 

(3) f''(x) = 0 has at least two roots in (0, 2) 

(4) f''(x) = 0 has 3 roots in (0, 2)

Answer 1

(3) f''(x) = 0 has at least two roots in (0, 2)