If `f(x) = x +1//2` Then. The number of real values of x for which the three unequal terms f(x) , f(2x) and f(4x) are in HP is

Options

A. 1
B. 0
C. 3
D. 2

Answer

a

Explanation

Given , `f(x) = x+1/2 = (2x+1)/2`
` :. F(2x)= 2x +1/2 rArr f(2x) = (4x+1)/2`
and ` f(4x) = 4x +1/2 rArr f(4x) = (8x+1)/2 `
Since, f(x) ,f(2x) and f(4x) are in HP.
` :. 1/(f(x)) , 1/(f(2x)) and 1/(f(4x))` are in HP.
` rArr 1/(f(2x))=(1/(f(x))+1/(f(4x)))/2`
` rArr 2/(4x+1) = (2/(2x+1)+2/(8x+1))/2`
` rArr 2/(4x+1)= (10x+2)/((2x+1)(8x+1))`
` rArr (2x+1) (8x+1) = (5x+1) (4x+1)`
`rArr 16x^(2) + 10x +1 = 20x^(2) + 9 x+1`
` rArr 4x^(2) - x = 0 `
` rArr x = 1/4 " " [ :' x != 0]`
Hence , there is one real value of x for which the three unequal terms are in HP.