The number of points, where the curve $$y = x^5 - 20x^3 + 50x + 2$$ crosses the x-axis, is ______.

We need to find the number of points where $$y = x^5 - 20x^3 + 50x + 2$$ crosses the x-axis.

Let $$f(x) = x^5 - 20x^3 + 50x + 2$$.

Find critical points.

$$f'(x) = 5x^4 - 60x^2 + 50 = 5(x^4 - 12x^2 + 10) = 0$$

Treating as a quadratic in $$x^2$$:

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

$$\sqrt{26} \approx 5.099$$

$$x^2 \approx 11.099 \text{ or } 0.901$$

Critical points: $$x \approx \pm 3.331$$ and $$x \approx \pm 0.949$$.

Evaluate $$f$$ at critical points.

$$f(-3.331) \approx 164.55 > 0$$

$$f(-0.949) \approx -29.13 < 0$$

$$f(0.949) \approx 33.13 > 0$$

$$f(3.331) \approx -160.55 < 0$$

Count sign changes using the Intermediate Value Theorem.

Since $$f$$ is continuous and changes sign between consecutive critical values and at the extremes:

1. $$f(x) \to -\infty$$ as $$x \to -\infty$$, and $$f(-3.331) > 0$$: one crossing in $$(-\infty, -3.331)$$.

2. $$f(-3.331) > 0$$ and $$f(-0.949) < 0$$: one crossing in $$(-3.331, -0.949)$$.

3. $$f(-0.949) < 0$$ and $$f(0.949) > 0$$: one crossing in $$(-0.949, 0.949)$$.

4. $$f(0.949) > 0$$ and $$f(3.331) < 0$$: one crossing in $$(0.949, 3.331)$$.

5. $$f(3.331) < 0$$ and $$f(x) \to +\infty$$ as $$x \to +\infty$$: one crossing in $$(3.331, +\infty)$$.

Each interval contains exactly one root (since $$f'$$ doesn't change sign within any interval between consecutive critical points), and each root is a simple root ($$f'$$ is nonzero there), so the curve truly crosses the x-axis at each.

The answer is $$5$$.