Question

Let $f\left(x\right)=x^{13}+x^{11}+x^9+x^7+x^5+x^3+x+19$. Then $f\left(x\right)=0$ is

• A. $13$ real roots
• B. Only one positive and only two negative real roots
• C. Not more than one real root
• D. Has two positive and one negative real root

Answer 1

The correct option is C Not more than one real root

$f\left(x\right)=x^{13}+x^{11}+x^9+x^7+x^5+x^3+x+19=0$
$f^{{}^{\prime }}\left(x\right)=13x^{12}+11x^{10}+9x^8+7x^6+5x^4+3x^2+1$
Since $f^{{}^{\prime }}\left(x\right)$consists of all even powers of $x$ therefore $f^{{}^{\prime }}\left(x\right)\ge 0$
That is $f\left(x\right)$ is an increasing function.Hence, it intersects the x-axis at only one point.
Therefore $f\left(x\right)$ has atmost one real root.