To determine the absolute maximum and minimum of the function $f(x) = (x - 3)^{\frac{4}{3}}$ over the interval $[-7, 7]$, we can follow these steps:

Step 1: Differentiate $f(x)$

1. Differentiate $f(x)$ to find $f'(x)$. $f(x) = (x - 3)^{\frac{4}{3}}$ Applying the power rule, we get: $f'(x) = \frac{4}{3} (x - 3)^{\frac{1}{3}}$

Step 2: Find Critical Points

1. Set $f'(x) = 0$: $\frac{4}{3} (x - 3)^{\frac{1}{3}} = 0$ Solving for $x$, we find that: $(x - 3)^{\frac{1}{3}} = 0 \Rightarrow x = 3$ So, $x = 3$ is a critical point.

Step 3: Evaluate $f(x)$ at the Endpoints and Critical Points

1. Calculate $f(-7)$: $f(-7) = (-7 - 3)^{\frac{4}{3}} = (-10)^{\frac{4}{3}} = 100^{\frac{2}{3}}$
2. Calculate $f(3)$: $f(3) = (3 - 3)^{\frac{4}{3}} = 0$
3. Calculate $f(7)$: $f(7) = (7 - 3)^{\frac{4}{3}} = 4^{\frac{4}{3}} = 16$

Step 4: Determine Maximum and Minimum Values

1. The values are:
   • $f(-7) = 100^{\frac{2}{3}}$
   • $f(3) = 0$
   • $f(7) = 16$
2. The absolute maximum value is $100^{\frac{2}{3}}$ at $x = -7$.
3. The absolute minimum value is $0$ at $x = 3$.

Conclusion

• Absolute Maximum: $f(x) = 100^{\frac{2}{3}}$ at $x = -7$
• Absolute Minimum: $f(x) = 0$ at $x = 3$

Would you like further details or explanations on any step?

Additional Questions to Explore

1. How would you compute $100^{\frac{2}{3}}$ exactly?
2. Why is $x = 3$ a minimum point instead of a maximum?
3. What happens to the function behavior if the exponent changes?
4. How do we know $x = 3$ is a critical point?
5. Can this function have more critical points?

Tip

Always check endpoints for absolute maximum and minimum values on closed intervals.