To solve the given problem, we need to analyze the intervals of increase and decrease for the polynomial function $f(x) = x^5 - 3x^3 + 4x$.

Step 1: Find the first derivative of $f(x)$

The first derivative, $f'(x)$, helps identify critical points and the behavior of $f(x)$: $f'(x) = 5x^4 - 9x^2 + 4$

Step 2: Solve $f'(x) = 0$ to find critical points

Set $f'(x) = 0$: $5x^4 - 9x^2 + 4 = 0$ Let $u = x^2$, so the equation becomes: $5u^2 - 9u + 4 = 0$ Solve using the quadratic formula: $u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \text{ where } a = 5, b = -9, c = 4$ $u = \frac{9 \pm \sqrt{(-9)^2 - 4(5)(4)}}{2(5)}$ $u = \frac{9 \pm \sqrt{81 - 80}}{10}$ $u = \frac{9 \pm 1}{10}$ $u = 1 \quad \text{or} \quad u = 0.8$ Thus, $x^2 = 1$ or $x^2 = 0.8$. This gives: $x = \pm 1, \quad x = \pm \sqrt{0.8}$

Step 3: Determine the sign of $f'(x)$ in each interval

The critical points divide the real number line into intervals: $(-\infty, -1)$, $(-1, -\sqrt{0.8})$, $(-\sqrt{0.8}, \sqrt{0.8})$, $(\sqrt{0.8}, 1)$, $(1, \infty)$. Test $f'(x)$ in each interval:

1. Interval $(-\infty, -1)$: Pick $x = -2$. Substitute into $f'(x)$: $f'(-2) = 5(-2)^4 - 9(-2)^2 + 4 = 80 - 36 + 4 = 48 > 0$ $f(x)$ is increasing.

2. Interval $(-1, -\sqrt{0.8})$: Pick $x = -0.9$. Substitute into $f'(x)$: $f'(-0.9) = 5(-0.9)^4 - 9(-0.9)^2 + 4$ Calculation shows $f'(-0.9) < 0$. $f(x)$ is decreasing.

3. Interval $(-\sqrt{0.8}, \sqrt{0.8})$: Pick $x = 0$. Substitute: $f'(0) = 5(0)^4 - 9(0)^2 + 4 = 4 > 0$ $f(x)$ is increasing.

4. Interval $(\sqrt{0.8}, 1)$: Pick $x = 0.9$. Substitute: Calculation shows $f'(0.9) < 0$. $f(x)$ is decreasing.

5. Interval $(1, \infty)$: Pick $x = 2$. Substitute: $f'(2) = 5(2)^4 - 9(2)^2 + 4 = 80 - 36 + 4 = 48 > 0$ $f(x)$ is increasing.

Step 4: Write the intervals

• $f(x)$ is increasing on $(-\infty, -1) \cup (-\sqrt{0.8}, \sqrt{0.8}) \cup (1, \infty)$.
• $f(x)$ is decreasing on $(-1, -\sqrt{0.8}) \cup (\sqrt{0.8}, 1)$.

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Related questions:

1. How do we find higher-order derivatives to classify critical points further?
2. What role does the second derivative play in determining concavity?
3. Can you use a graphing utility to verify these intervals visually?
4. How can you confirm the nature of turning points using $f''(x)$?
5. What happens to $f(x)$ as $x \to \pm\infty$?

Tip: Always test critical points carefully to avoid algebraic errors when solving for intervals.