q) A function f such that f '(a) = f '' (a) = f ' ' '(a).........=f²ⁿ(a)=0

   and f has a local maximum at value of b at x=a

  find f(x).

Please answer this with proper explanation.

It has to be e^x. It is the only function (I know of) whose any derivative is equal to the function

Nope...

The answer is b- (x-a)²ⁿ⁺²

but i dont understand how to get the answer.

Some1 else try plz

THE  QUESTION  HAS  BEEN  ANSWERED  WITHOUT  USING  ANY  SPECIAL  METHODS.  I  JUST  CIRCULATED  THROUGH  THE  COMMON  TYPE  OF  FUNCTIONS:

1.  POLYNOMIAL

2.  TRIGONOMETRIC

3.  LOGARITHMIC

4.  EXPONENTIAL        etc.......  SO THAT THE GIVEN CONDITIONS ARE SATISFIED............

THEREFORE

f (x)   =   - [ log ( x - a + 1 ) ] ^{2n + m}   +   b       where 'm' is any even natural number.

f (x)   =   - ( x - a )^{2n + m}   +  b      where 'm' is any even natural number.