Let's work through the problem in the image step-by-step. Here’s a breakdown of each part:

Given Information

We are given the function: $f(x) = \sqrt{1 - x^3}$

Problem Breakdown

1. Verify that $D_f = ] -\infty , 1]$ and then calculate $f\left( \sqrt[3]{\frac{5}{9}} \right)$.

To determine the domain of $f(x) = \sqrt{1 - x^3}$, we need: $1 - x^3 \geq 0$ Since $f(x)$ involves a square root, the expression inside must be non-negative. We can solve the inequality $1 - x^3 \geq 0$ to find the allowed values of $x$.

Once the domain is confirmed, we calculate $f\left( \sqrt[3]{\frac{5}{9}} \right)$.

2. Calculate $\lim_{x \to 1^-} \frac{f(x)}{x - 1}$ and interpret the geometric meaning of the result.

Here, we need to calculate the limit as $x$ approaches 1 from the left of $\frac{f(x)}{x - 1}$. This limit may give insight into the behavior of $f(x)$ near $x = 1$, possibly indicating if there’s a vertical asymptote or other significant geometric property.

3. (a) Calculate $f'(x)$ for all $x$ in $] -\infty , 1]$ and deduce the variations of $f$.

(b) Show that $f$ has an inverse function $f^{-1}$ defined on a certain interval $J$. (c) Calculate $\sqrt{1 - (f^{-1}(\pi))^3}$ and $\left( f^{-1} \right) \left( \frac{2}{3} \right)$.

1. Find $f'(x)$ by differentiating $f(x)$.
2. Determine if $f(x)$ is monotonic (increasing or decreasing) on its domain, as this will help in identifying the possibility of an inverse function.
3. For (c), further calculations are needed based on the inverse function properties.

Would you like me to proceed with detailed solutions for each part? Let me know if you have any specific area where you’d like more focus.

Follow-up Questions

1. How can we solve the inequality $1 - x^3 \geq 0$?
2. What does the limit $\lim_{x \to 1^-} \frac{f(x)}{x - 1}$ tell us about $f(x)$ near $x = 1$?
3. What techniques can we use to find the derivative $f'(x)$ of a function involving a square root?
4. Why is the monotonicity of $f(x)$ important in determining the existence of an inverse function?
5. How would we approach calculating $\sqrt{1 - (f^{-1}(\pi))^3}$ without knowing the explicit form of $f^{-1}(x)$?

Tip

When dealing with square roots in functions, always pay close attention to the domain since it directly affects the validity of your solutions.