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