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