The number of real solutions of x^{7}+5x^{3}+ | vedclass

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(B) Let $f(x) = x^{7} + 5x^{3} + 3x + 1$.Find the derivative of the function:$f'(x) = 7x^{6} + 15x^{2} + 3$.Since $x^{6} \ge 0$ and $x^{2} \ge 0$ for

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$1$

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(B) Let $f(x) = x^{7} + 5x^{3} + 3x + 1$.Find the derivative of the function:$f'(x) = 7x^{6} + 15x^{2} + 3$.Since $x^{6} \ge 0$ and $x^{2} \ge 0$ for all real $x$,it follows that $7x^{6} \ge 0$ and $15x^{2} \ge 0$.Therefore,$f'(x) = 7x^{6} + 15x^{2} + 3 \ge 3 > 0$ for all $x \in \mathbb{R}$.Since $f'(x) > 0$ for all $x$,the function $f(x)$ is a strictly increasing function.As $x 	o -\infty$,$f(x) 	o -\infty$,and as $x 	o \infty$,$f(x) 	o \infty$.Since $f(x)$ is a continuous and strictly increasing function that ranges from $-\infty$ to $\infty$,by the Intermediate Value Theorem,it must cross the $x$-axis exactly once.Thus,the number of real solutions is $1$.

The number of real solutions of $x^{7}+5x^{3}+3x+1=0$ is equal to............

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