The roots of `(x-41)^(49)+(x-49)^(41)+(x-2009)^(2009)=0` are

Options

A. all necessarily real
B. non-real except one positive real root
C. non-real except three positive real roots
D. non-real except for three real roots of which exactly one is positive

Answer

B

Explanation

We have `(x-41)^(49)+(x-49)^(41)+(x-2009)^(2009)=0`
Let `f(x)=(x-41)^(49)+(x-49)^(41)+(x-2009)^(2009)`
`therefore" "f'(x)=49(x-41)^(48)+41(x-49)^(40)+2009(x-2009)^(48)gt0`
Hence f(x) will cut x-axis only once `rArr" 1 real root"`