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

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(B) Given $f(x) = x^{13} + x^{11} + x^{9} + x^{7} + x^{5} + x^{3} + x + 12$. Taking the derivative, we get $f'(x) = 13x^{12} + 11x^{10} + 9x^{8} + 7x^

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$f(x)$ has exactly one real root

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(B) Given $f(x) = x^{13} + x^{11} + x^{9} + x^{7} + x^{5} + x^{3} + x + 12$. Taking the derivative, we get $f'(x) = 13x^{12} + 11x^{10} + 9x^{8} + 7x^{6} + 5x^{4} + 3x^{2} + 1$. Since $x^{2n} \ge 0$ for all $x \in \mathbb{R}$, it follows that $f'(x) > 0$ for all $x \in \mathbb{R}$. This implies that $f(x)$ is a strictly increasing function. As $x 	o \infty$, $f(x) 	o \infty$ and as $x 	o -\infty$, $f(x) 	o -\infty$. By the Intermediate Value Theorem, since $f(x)$ is continuous and strictly increasing, it must cross the $x$-axis exactly once. Therefore, $f(x) = 0$ has exactly one real root.

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

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