Question

Let $f\left(x\right)$ be a monotonic polynomial of $2m-2$ degree where $m\in N$ then the equation $f\left(x\right)+f\left(3x\right)+f\left(5x\right)+..+f\left((2m-1)x\right)=2m-1$ has

• A. atleast one real root
• B. $(2m-1)$ real roots
• C. exactly one real root
• D. none of these

Answer 1

The correct option is A atleast one real root

$f\left(x\right)$ is monotonic $\Rightarrow f^{\prime }\left(x\right)<0$ or $f^{\prime }\left(x\right)>0\forall x\in R$

$\Rightarrow f^{\prime }\left(px\right)<0$ or $f^{\prime }\left(px\right)>0\forall x\in R$

$\Rightarrow f\left(px\right)$ is also montonic

$\Rightarrow f\left(x\right)+f\left(3x\right)+f\left(5x\right)+..+f\left((2m-1)x\right)=2m-1$ is a monotonic polynomial of odd degree $(2m-1),$ So it will attain all real values only once.