To find the absolute maximum and minimum of the function $f(x) = x^{4/5}$ on the interval $[-1, 2]$, we need to follow these steps:

Step 1: Analyze the function

The function $f(x) = x^{4/5}$ is continuous, but it has a point of non-differentiability at $x = 0$, as the derivative involves fractional exponents.

Step 2: Compute the derivative

To find the critical points, we calculate the derivative:

$f'(x) = \frac{4}{5}x^{-1/5} = \frac{4}{5} \cdot \frac{1}{x^{1/5}}$

This derivative is undefined at $x = 0$, so we have a critical point there.

Step 3: Evaluate endpoints and critical points

We now evaluate $f(x)$ at the endpoints of the interval and at the critical point $x = 0$.

• $f(-1) = (-1)^{4/5} = 1$
• $f(0) = 0^{4/5} = 0$
• $f(2) = 2^{4/5} \approx 1.31951$

Step 4: Conclusion

• The absolute minimum value occurs at $x = 0$ and is $\text{min} = 0$.
• The absolute maximum value occurs at $x = 2$ and is $\text{max} = 2^{4/5} \approx 1.31951$.

Would you like further details on any part of this solution?

Here are five related questions:

1. How do you determine where a function is non-differentiable?
2. How do fractional exponents affect the shape of a function's graph?
3. Can a non-differentiable point be a minimum or maximum?
4. What is the difference between local and absolute extrema?
5. How do you evaluate complex fractional exponents?

Tip: Always evaluate both critical points and endpoints when finding absolute extrema on a closed interval.