The number of distinct real roots of the equa | vedclass

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(D) Let $f(x) = x^{7}-7x$. We want to find the number of real roots of $f(x) = 2$.First,find the derivative: $f'(x) = 7x^{6}-7 = 7(x^{6}-1) = 7(x^{3}-

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$3$

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(D) Let $f(x) = x^{7}-7x$. We want to find the number of real roots of $f(x) = 2$.First,find the derivative: $f'(x) = 7x^{6}-7 = 7(x^{6}-1) = 7(x^{3}-1)(x^{3}+1) = 7(x-1)(x^{2}+x+1)(x+1)(x^{2}-x+1)$.The critical points are $x = 1$ and $x = -1$.Evaluate $f(x)$ at these points:$f(1) = 1^{7}-7(1) = 1-7 = -6$.$f(-1) = (-1)^{7}-7(-1) = -1+7 = 6$.As $x 	o \infty$,$f(x) 	o \infty$,and as $x 	o -\infty$,$f(x) 	o -\infty$.The function increases on $(-\infty, -1)$,decreases on $(-1, 1)$,and increases on $(1, \infty)$.Since $f(-1) = 6 > 2$ and $f(1) = -6 Thus,there are $3$ distinct real roots.

The number of distinct real roots of the equation $x^{7}-7x-2=0$ is

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