We are given the function $f(x) = 2x - 4 \sin(x)$ over the interval $0 \leq x \leq 2\pi$, and we need to analyze its behavior (intercepts, relative extrema, points of inflection, and asymptotes).

1. Intercepts

x-intercepts: To find the x-intercepts, set $f(x) = 0$: $2x - 4 \sin(x) = 0$ $x = 2 \sin(x)$ We need to solve this equation numerically for $x \in [0, 2\pi]$.

y-intercept: The y-intercept occurs when $x = 0$: $f(0) = 2(0) - 4\sin(0) = 0$ Thus, the y-intercept is $(0, 0)$.

2. Relative Extrema (Maxima and Minima)

To find the relative extrema, we first find the critical points by taking the derivative and setting it equal to zero.

$f'(x) = 2 - 4 \cos(x)$ Set $f'(x) = 0$: $2 - 4 \cos(x) = 0$ $\cos(x) = \frac{1}{2}$ The solutions to $\cos(x) = \frac{1}{2}$ are $x = \frac{\pi}{3}$ and $x = \frac{5\pi}{3}$ within the interval $[0, 2\pi]$.

To classify these points as minima or maxima, use the second derivative test.

$f''(x) = 4 \sin(x)$ Evaluate $f''(x)$ at the critical points:

• At $x = \frac{\pi}{3}$, $\sin\left(\frac{\pi}{3}\right) > 0$, so this is a local minimum.
• At $x = \frac{5\pi}{3}$, $\sin\left(\frac{5\pi}{3}\right) < 0$, so this is a local maximum.

3. Points of Inflection

Points of inflection occur when the second derivative changes sign, i.e., when $f''(x) = 0$: $f''(x) = 4 \sin(x) = 0$ $\sin(x) = 0$ The solutions are $x = 0, \pi, 2\pi$.

Evaluate the change in concavity around these points. At these points, the second derivative changes sign, indicating inflection points.

4. Asymptotes

Since the function $f(x) = 2x - 4 \sin(x)$ is a polynomial combined with a sine function, it does not have any vertical or horizontal asymptotes.

5. Summary

• Intercepts: $(0, 0), (\text{Solve } 2x = 4 \sin(x) \text{ numerically for another intercept in } [0, 2\pi])$

• Relative Minimum: $\left( \frac{\pi}{3}, f\left( \frac{\pi}{3} \right) \right)$

• Relative Maximum: $\left( \frac{5\pi}{3}, f\left( \frac{5\pi}{3} \right) \right)$

• Points of Inflection: $(0, f(0)), (\pi, f(\pi)), (2\pi, f(2\pi))$

• Asymptotes: DNE (no asymptotes).

Would you like me to provide a detailed graph of the function or help with solving the intercept equation numerically?

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Related questions:

1. How do you solve $2x = 4\sin(x)$ numerically?
2. Can you explain how to apply the second derivative test?
3. Why are there no asymptotes for this function?
4. How do you determine points of inflection using the second derivative?
5. What is the significance of relative extrema in a function's graph?

Tip:

Always verify relative extrema by using both the first and second derivative tests to ensure accuracy in classification!