Let f(x)=x^{13}+x^{11}+x^{9}+x^{7}+x^{5}+x^{3 | vedclass

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(C) Given $f(x) = x^{13} + x^{11} + x^{9} + x^{7} + x^{5} + x^{3} + x + 19$. Taking the derivative with respect to $x$: $f'(x) = 13x^{12} + 11x^{10} +

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not more than one real root

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(C) Given $f(x) = x^{13} + x^{11} + x^{9} + x^{7} + x^{5} + x^{3} + x + 19$. Taking the derivative with respect to $x$: $f'(x) = 13x^{12} + 11x^{10} + 9x^{8} + 7x^{6} + 5x^{4} + 3x^{2} + 1$. Since all exponents of $x$ in $f'(x)$ are even and the coefficients are positive, $f'(x) \geq 1$ for all real $x$. Thus, $f'(x) > 0$ for all $x \in \mathbb{R}$, which means $f(x)$ is a strictly increasing function. $A$ strictly increasing function can intersect the $x$-axis at most once. Therefore, $f(x) = 0$ has not more than one real root.

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

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