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### (a) Critical Points
The first derivative of the function \( f(x) = \frac{3}{4}(x^2 - 1)^{\frac{2}{3}} \) is:
\[
f'(x) = x(x^2 - 1)^{-\frac{1}{3}}
\]
Critical points occur when \( f'(x) = 0 \) or when \( f'(x) \) is undefined:
- \( f'(x) = 0 \) gives a critical point at \( x = 0 \).
- \( f'(x) \) is undefined when \( x = \pm 1 \), which correspond to points where the original function has a non-differentiable cusp.

Now let's move to the next parts (b) through (e).### (b) Intervals of Increase and Decrease:
By analyzing the sign of the derivative in different intervals:
- In the interval \( (-\infty, -1) \), \( f'(x) \) is negative (\( -1.387 \)), so the function is **decreasing**.
- In the interval \( (-1, 0) \), \( f'(x) \) has complex values, which suggests that the behavior near \( x = -1 \) is undefined due to the cusp.
- In the interval \( (0, 1) \), \( f'(x) \) also has complex values, indicating the same undefined behavior near \( x = 1 \).
- In the interval \( (1, \infty) \), \( f'(x) \) is positive (\( 1.387 \)), so the function is **increasing**.

Thus, the function is decreasing on \( (-\infty, -1) \) and increasing on \( (1, \infty) \). The points \( x = -1 \) and \( x = 1 \) are undefined due to the function's non-differentiability at these points.

Now let's move on to local maxima and minima in part (c).### (c) Local Maxima and Minima:
- The function has **no local maxima** in the valid intervals.
- The **local minimum** occurs at \( x = 0 \), but the value of the function at this point is complex: \( f(0) = -0.375 + 0.649i \). This suggests that the function exhibits complex behavior at \( x = 0 \), potentially due to the fractional exponent.

Next, let's proceed to finding the inflection points (part d) and discussing concavity (part e).It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like to explore the concavity and inflection points manually, you can differentiate the function once more to obtain the second derivative and solve for where it's zero or undefined.

Do you want any further clarification on the earlier parts? Feel free to ask!

Here are 5 related questions to consider:
1. What is the geometric significance of critical points in graphing?
2. How do cusps affect the behavior of a function like this one?
3. Can you explain how complex values arise in this context?
4. How does fractional exponents influence the smoothness of curves?
5. What conditions lead to a function having inflection points?

**Tip**: When studying critical points, always check the domain and differentiability to avoid misleading results.

For the function f(x) = 3/4 * (x^2 - 1)^(2/3), find the critical points, identify the intervals on which the function is increasing and decreasing, obtain local maxima and minima, identify inflection points, and discuss the concavity of the curve.

Solve the calculus problem for the function f(x) = 3/4 * (x^2 - 1)^(2/3) by finding the critical points, determining the intervals of increase and decrease, identifying local maxima and minima, and discussing the concavity and inflection points. This step-by-step solution covers key calculus concepts suitable for Grades 11-12 and college-level students.

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