The number of distinct real roots of the equation x ^ 7 - 7x - 2 =...

Given,

${x}^{7} - 7x - 2 = 0$

${x}^{7} - 7x = 2$

Now taking $f\left(x\right) = {x}^{7} - 7x$ (odd) & $y = 2$

Now differentiating $f\left(x\right)$ w.r.t $x$ we get,

${f}^{'}\left(x\right) = 7\left({x}^{6} - 1\right) = 7\left({x}^{2} - 1\right)\left({x}^{4} + {x}^{2} + 1\right)$

${f}^{'}\left(x\right) = 0 ⇒ x = \pm 1$

So points of extrema are $- 1 \text{ and } 1$, Now plotting the graph to check number of solution we get,

$f\left(x\right) = 2$ has $3$ real distinct solution.