To determine where the function $f(x) = x + 12x^{2/3}$ is increasing and decreasing, we need to find its first derivative and analyze the sign of the derivative.

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

$f(x) = x + 12x^{2/3}$

Using the power rule for differentiation:

$f'(x) = \frac{d}{dx}(x) + \frac{d}{dx}(12x^{2/3})$

$f'(x) = 1 + 12 \cdot \frac{2}{3}x^{-1/3}$

$f'(x) = 1 + 8x^{-1/3}$

2. Simplify the first derivative:

$f'(x) = 1 + 8x^{-1/3}$

$f'(x) = 1 + \frac{8}{x^{1/3}}$

3. Determine where the derivative is positive or negative:

The function $f(x)$ is increasing where $f'(x) > 0$ and decreasing where $f'(x) < 0$.

To find the critical points, set the first derivative equal to zero:

$f'(x) = 0$

$1 + \frac{8}{x^{1/3}} = 0$

$\frac{8}{x^{1/3}} = -1$

$8 = -x^{1/3}$

$x^{1/3} = -8$

$x = (-8)^3$

$x = -512$

4. Analyze the sign of the derivative around the critical point $x = -512$:

• For $x < -512$: $x = -1000$ $f'(-1000) = 1 + \frac{8}{(-1000)^{1/3}}$ $f'(-1000) = 1 + \frac{8}{-10}$ $f'(-1000) = 1 - 0.8$ $f'(-1000) = 0.2$ Thus, $f'(x) > 0$ for $x < -512$, so $f(x)$ is increasing.

• For $x > -512$: $x = 0$ $f'(0) = 1 + \frac{8}{0}$ The expression $\frac{8}{0}$ is undefined, indicating that there is a vertical asymptote at $x = 0$.

So, we consider values closer but greater than $-512$: $x = -1$ $f'(-1) = 1 + \frac{8}{(-1)^{1/3}}$ $f'(-1) = 1 + 8(-1)$ $f'(-1) = 1 - 8$ $f'(-1) = -7$ Thus, $f'(x) < 0$ for $x > -512$, so $f(x)$ is decreasing.

5. Summary:

• The function $f(x) = x + 12x^{2/3}$ is increasing for $x < -512$.
• The function $f(x) = x + 12x^{2/3}$ is decreasing for $x > -512$.

Would you like further details or have any questions?

Here are 8 related questions that might interest you:

1. How do you find the second derivative of a function?
2. What are critical points, and why are they important?
3. How can you determine concavity and points of inflection for a function?
4. What is the importance of the first derivative in analyzing a function's behavior?
5. Can a function be increasing and decreasing at the same point?
6. How do you handle vertical asymptotes in derivative analysis?
7. What is the difference between local and absolute extrema?
8. How can you apply the first derivative test to determine local extrema?

Tip: Always check the domain of the function before analyzing its derivatives, as undefined points can affect the behavior of the function significantly.