The number of points, where the curve $y=x^5-20x^3+50x+2$ crosses the $x$-axis, is _____.
Step-by-Step Solution
Key Concept: Compute $f'(x)=5x^4-60x^2+50=5(x^4-12x^2+10)$ and find its roots to determine local extrema, then check sign changes of $f$ around those roots.
$f'(x)=5(x^4-12x^2+10)=0\Rightarrow x^2=6\pm\sqrt{26}$. Critical points near $x\approx\pm3.31,\pm0.95$. Checking: $f(0)=2>0,\ f(1)>0,\ f(2)<0,\ f(-1)<0,\ f(-2)>0,\ f(4)>0,\ f(-4)<0$. The curve crosses the $x$-axis $\boxed{5}$ times.
Correct Answer: 5