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If f(x) is a polynomial and f:R->R such that f(2x)=f'(x).f''(x). Find f(3).

If f(x) is a polynomial and f:R->R such that f(2x)=f'(x).f''(x). Find f(3).

Grade:12

2 Answers

Ajay
209 Points
7 years ago
Here is my attempt to solution, let me know if it is correct........................................................................
Given\quad f(2x)\quad =\quad { f }^{ ` }(x){ f }^{ `` }(x)\\ This\quad only\quad possible\quad if\quad f(x)\quad is\quad a\quad polynomial\quad of\quad degree\quad 3\\ let\quad f(x)\quad =\quad { a }_{ 0 }+{ a }_{ 1 }x+{ a }_{ 2 }{ x }^{ 2 }+{ a }_{ 3 }{ x }^{ 3 }\\ f(2x)\quad =\quad { a }_{ 0 }+2{ a }_{ 1 }x+{ 4a }_{ 2 }{ x }^{ 2 }+8{ a }_{ 3 }{ x }^{ 3 }\quad ...............................(1)\\ { f }^{ ` }(x){ f }^{ `` }(x)\quad =\quad ({ a }_{ 1 }+2{ a }_{ 2 }{ x }+{ 3a }_{ 3 }{ x }^{ 2 })(2{ a }_{ 2 }+{ 6a }_{ 3 }{ x })..............(2)\\ comparing\quad coeficients\quad and\quad solving\quad we\quad get\\ { a }_{ 0 }\quad =\quad { a }_{ 1 }\quad =\quad { a }_{ 2 }\quad =\quad 0\quad and\quad { a }_{ 3 }\quad =4/9\\ Hence\quad f(x)\quad =\quad 4/9{ x }^{ 3 }\\ f(3)\quad =\quad 12
Kumar
37 Points
7 years ago
Suppose that f(x) is an infinite degree polynomial i.e. f(x)=exp(x),  and satiesfied the given condition f(2x)=f’(x)f”(x). Thus the value of f(3)=exp(3).

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