Question

If $f\left(x\right)$ is a polynomial of degree $n(>2)$ and $f\left(x\right)=f(k-x),($ where $k$ is a fixed real number$),$ then degree of $f^{\prime }\left(x\right)$ is

• A. $n$
• B. $n-1$
• C. $n-2$
• D. None of these

Answer 1

The correct option is B $n-1$

Clearly, $f\left(x\right)$ must be of the from

$f\left(x\right)=a_0[x^n+{(k-x)}^n]+a_1[x^{n-1}+{(k-x)}^{n-1}]+...+a_{n-1}[x+(k-x)]+a_{n.}$

It may be noted that $n$ must be even for otherwise $f\left(x\right)$ will become a polynomial of degree $n-1.$

Clearly, $f^{\prime }\left(x\right)$ is a polynomial of degree $n-1.$