Question

The roots of $(x-41)^{49}+(x-49)^{41}+(x-2009)^{2009}=0$ are

• 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 1

Answer: (B)

We have,
$(x-41)^{49}+(x-49)^{41}+(x-2009)^{2009}=0$
Let $f(x)=(x-41)^{49}+(x-49)^{41}+(x-2009)^{2009}$
$\Rightarrow f'(x) =49(x-41)^{48}+41(x-49)^{40}+2009(x-2009)^{2008}$
Here, $f'(x) >\,0, \forall\, x \, \in R$
$\therefore f (x)$ is monotonic increasing function
Hence, $f (x)$ has only one real root