IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

mona840

The function f(x) satisfies f( x+p) + f(x-p) = sqrt(3)f(x) , xE R , p>0

Prove tat f(x) is periodic n find its period

ankurgupta91

f(x+p) +f(x-p) = sqrt(3)f(x)

nw put x= x+p
f(x+2p) +f(x) = sqrt(3)f(x+p) -------(1)

nw put x= x-p
f(x) +f(x-2p) = sqrt(3)f(x-p) ---------(2)

add (1) and (2)
f(x+2p) + 2f(x) +f(x-2p) = sqrt(3) [ f(x+p) +f(x-p)]
f(x+2p) + 2f(x) +f(x-2p) = sqrt(3) sqrt(3) f(x)

f(x+2p) +f(x-2p) = f(x) -------(3)
nw put x = x+2p
f(x+4p) + f(x) = f(x+2p) --------(4)

add(3) and (4)
f(x+4p) +f(x-2p) =0

nw put x = x+2p
f(x+6p) +f(x) =0 --------(5)

nw put x= x -4p
f(x) +f(x-6p) =0 --------(6)

subtract 5 and 6

f(x+6p) = f(x-6p)
nw put x = x+6p
f(x) = f(x+12p)

so the period is 12p ............
thats the answer......

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