The number of points, where the curve y = x ^ 5 - 20 x ^ 3 + 50x +...

Let, $f ( x ) = {x}^{5} - 20{x}^{3} + 50x + 2$

Now differentiating the function $f$\left(x\right)$with respect to$x$we get,</p><p>${f}^{'} ( x ) = 5{x}^{4} - 60{x}^{2} + 50$</p><p>$⇒ {f}^{'} ( x ) = 5\left({x}^{4} - 12{x}^{2} + 10\right)$</p><p>Now equating${f}^{'} ( x ) = 0$we get,</p><p>${x}^{4} - 12{x}^{2} + 10 = 0$</p><p>$⇒ {x}^{2} = \dfrac{12 \pm \sqrt{144 - 40}}{2}$</p><p>$⇒ {x}^{2} = 6 \pm \sqrt{26}$</p><p>$⇒ {x}^{2} \approx 11 . 1, 0 . 9$</p><p>$⇒ x \approx \pm 3 . 31, \pm 0 . 95$</p><p>Now finding the nature of function at integer value near by above value we get,</p><p>$f\left(0\right) = 2, f\left(1\right) > 0, f\left(2\right) 0$</p><p>Now plotting the graph by observing above values, we get,</p><p><img src="https://cdn-question-pool.Quantrex.app/pyq/jee_main/4bbf06d4-950b-4ae8-99cf-74634038d164.png" style="width: 299px; height: 146px;" /></p><p>Hence, it will cross$x -$axis$5 times.