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

`(x-41)^(49) + (x-49)^(41) + (x-2009)^(2009)=0`
`f(x) = (x-41)^(49)+(x-49)^(41)+(x-2009)^(2009)`
`f'(x) = 49(x-41)^(48) + 41(x-49)^(40) + 2009(x-2009)^(48) gt 0`
hence ` f(x)` will cut x-axis only once.
1 real root.