1. **State the problem:** We are given the function $f(x) = x \sqrt{5 - 2x}$ and want to understand its behavior. 2. **Domain:** The expression inside the square root must be non-negative for real values, so: $$5 - 2x \geq 0$$ 3. **Solve the inequality:** $$5 \geq 2x$$ $$\frac{5}{2} \geq x$$ 4. **Domain conclusion:** The domain is all $x$ such that: $$x \leq \frac{5}{2}$$ 5. **Function behavior:** The function is defined for $x \leq 2.5$ and involves a product of $x$ and the square root term. 6. **Derivative (optional for analysis):** To find critical points, use the product and chain rules: $$f(x) = x (5 - 2x)^{1/2}$$ $$f'(x) = (5 - 2x)^{1/2} + x \cdot \frac{1}{2} (5 - 2x)^{-1/2} (-2)$$ $$= (5 - 2x)^{1/2} - \frac{x}{(5 - 2x)^{1/2}}$$ 7. **Simplify derivative:** $$f'(x) = \frac{(5 - 2x) - x}{(5 - 2x)^{1/2}} = \frac{5 - 3x}{(5 - 2x)^{1/2}}$$ 8. **Critical points:** Set numerator to zero: $$5 - 3x = 0$$ $$x = \frac{5}{3}$$ 9. **Summary:** - Domain: $(-\infty, \frac{5}{2}]$ - Critical point at $x = \frac{5}{3}$ - Function involves $x$ times the square root of $5 - 2x$. This analysis helps understand where the function is defined and where it may have maxima or minima.