The number of points, where the curve y = x 5 - 20 x 3 + 50 x + 2 crosses the x -axis, is __________ . [2023]

Question

The number of points, where the curve $y = x^{5} - 20 x^{3} + 50 x + 2$ crosses the $x$ -axis, is __________ . [2023]

Smart Achievers · Accepted Answer

(5)

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

$\Rightarrow \frac{d y}{d x} = 5 x^{4} - 60 x^{2} + 50 = 5 (x^{4} - 12 x^{2} + 10)$

$Put \frac{d y}{d x} = 0 \Rightarrow x^{4} - 12 x^{2} + 10 = 0$

$\Rightarrow x^{2} = \frac{12 \pm \sqrt{144 - 40}}{2} \Rightarrow x^{2} = 6 \pm \sqrt{26} \Rightarrow x^{2} \approx 6 \pm 5.1$

$\Rightarrow x^{2} \approx 11.1, 0.9 \Rightarrow x \approx \pm 3.3, \pm 0.95$

$f (0) = 2, f (1) = + v e, f (2) = - v e; f (- 1) = - v e, f (- 2) = + v e$

$∴$ $y crosses the x -axis at 5 points.$

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