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Let f:R to R satisfy f(x+y) =2^(x) f(y) ...

Let `f:R to R` satisfy `f(x+y) =2^(x) f(y) +4^(y)f(x) , AA x , y in R `. If f(2) = 3 , then 14. `(f'(4))/(f'(2))` is equal to ______.

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Knowledge Check

  • If f(x+y) = f(x) + f(y) + |x|y+xy^(2),AA x, y in R and f'(0) = 0 , then

    A
    f need not be differentiable at every non-zero x
    B
    f is differentiable for all `x in R`
    C
    f is twice differentiable at x = 0
    D
    None of the above

  • Let f:R to R such that f(x+y)+f(x-y)=2f(x)f(y) for all x,y in R . Then,

    A
    f(x) an even function , if `f(0) ne 0`
    B
    f(x) is an odd function, if `f(0) ne 0`
    C
    f(x) an even function , if `f(0)=0`
    D
    f(x) is an odd function , if `f(0)=0`

  • If f:R to R such that f (x + y) + f (x - y) =2f (x) f (y) AA x,y in R and f (0) ne 0 , then f (x) is an :

    A
    even function
    B
    odd function
    C
    periodic function
    D
    none