Question

Let $f\left(x\right)=x^{13}+x^{11}+x^9+x^7+x^5+x^3+x+12.$ Then

• A. $f\left(x\right)$ has $13$ non-zero real roots
• B. $f\left(x\right)$ has exactly one real root
• C. $f\left(x\right)$ has exactly one pair of imaginary roots
• D. $f\left(x\right)$ has no real root

Answer 1

The correct option is B $f\left(x\right)$ has exactly one real root

$f\left(x\right)=x^{13}+x^{11}+x^9+x^7+x^5+x^3+x+12f^{\prime }\left(x\right)=13x^{12}+11x^{10}+9x^8+7x^6+5x^4+3x^2+1>0\forall x\in R$
i.e Monotonically increasing $\forall x\in R$.
$\Rightarrow f\left(x\right)$ intersects $x-$axis at only one point
$\therefore$ exactly one solution.