JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Application of Derivatives

Differentiating, $y^{\prime }=5x^4-60x^2+50=5\left(x^4-12x^2+10\right)$, which vanishes at $x^2=6\pm \sqrt{26}$, giving four turning points at $x\approx -3.3,-0.95,0.95,3.3$. Evaluating the polynomial at these points gives approximately $+164$ at $x\approx -3.3$, $-29$ at $x\approx -0.95$, $+33$ at $x\approx 0.95$ and $-160$ at $x\approx 3.3$. Since the curve tends to $-\infty$ as $x\to -\infty$ and to $+\infty$ as $x\to +\infty$, it rises to a positive maximum, falls to a negative minimum, rises to a positive maximum, falls to a negative minimum, and finally increases without bound. The value changes sign across each of the five monotone stretches, so the curve meets the axis once in each. The curve crosses the $x$-axis at $5$ points.