The number of real roots of the equation x^5+ | vedclass

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(A) Let $f(x) = x^5+3x^3+4x+30$. Taking the derivative with respect to $x$,we get $f'(x) = 5x^4+9x^2+4$. Since $x^4 \ge 0$ and $x^2 \ge 0$ for all $x

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$1$

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(A) Let $f(x) = x^5+3x^3+4x+30$. Taking the derivative with respect to $x$,we get $f'(x) = 5x^4+9x^2+4$. Since $x^4 \ge 0$ and $x^2 \ge 0$ for all $x \in \mathbb{R}$,it follows that $5x^4+9x^2+4 \ge 4 > 0$ for all $x \in \mathbb{R}$. Since $f'(x) > 0$ for all $x \in \mathbb{R}$,the function $f(x)$ is strictly increasing on its domain. $A$ strictly increasing continuous function can cross the $x$-axis at most once. Since $\lim_{x 	o -\infty} f(x) = -\infty$ and $\lim_{x 	o \infty} f(x) = \infty$,by the Intermediate Value Theorem,there exists exactly one real root.

The number of real roots of the equation $x^5+3x^3+4x+30=0$ is

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