Let f be a differentiable real valued function satisfying f(x+y) = f(x) + f(2y) + 6xy(x+2y) for all x, y in R. Then prove that f"(0), f"(1),f"(2),... are in AP.

Dear student,
fx+y = fx+f2y+6xyx+2yput y=xf2x = fx+f2x+6x2x+2xfx=-18x3differentiatef'x=-54x2differentiatef''x = -108xf''0 = 0f''1 = -108f''2 = -108×2then 2f''1 =f''0+f''2 so  f''1 ,f''0,f''2  are in AP
Regards

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