Question

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

• 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

Answer: (C)

We have,
$f(x)=x^{13}+x^{11}+x^9+x^7+x^5+x^3+x+19$
$\Rightarrow f^{\prime }(x)=13x^{12}+11x^{10}+9x^8$
$+7x^6+5x^4+3x^2+1$
$\therefore f^{\prime }(x)$ has no real root.
$\therefore f(x)=0$ has not more than one real root.