b. \( y=\frac{\left(x^{2}+5\right) \sqrt{5 x-3}}{3 x+5} \)

To analyze the function \( y = \frac{(x^2 + 5) \sqrt{5x - 3}}{3x + 5} \), we can consider several aspects such as its domain, behavior, and any critical points. ### 1. **Domain** The domain of the function is determined by the conditions under which the expression is defined: - The square root \( \sqrt{5x - 3} \) requires that \( 5x - 3 \geq 0 \), which simplifies to: \[ x \geq \frac{3}{5} \] - The denominator \( 3x + 5 \) must not be zero: \[ 3x + 5 \neq 0 \implies x \neq -\frac{5}{3} \] However, since \( -\frac{5}{3} < \frac{3}{5} \), this condition does not affect the domain. Thus, the domain of \( y \) is: \[ x \in \left[\frac{3}{5}, \infty\right) \] ### 2. **Behavior at the endpoints of the domain** - As \( x \to \frac{3}{5} \): \[ y = \frac{\left(\left(\frac{3}{5}\right)^2 + 5\right) \sqrt{5 \cdot \frac{3}{5} - 3}}{3 \cdot \frac{3}{5} + 5} = \frac{\left(\frac{9}{25} + 5\right) \sqrt{0}}{3 \cdot \frac{3}{5} + 5} = 0 \] - As \( x \to \infty \): The leading terms dominate: \[ y \approx \frac{x^2 \sqrt{5x}}{3x} = \frac{x^2 \sqrt{5} \sqrt{x}}{3x} = \frac{\sqrt{5} x^{3/2}}{3} \] Thus, \( y \to \infty \) as \( x \to \infty \). ### 3. **Critical Points** To find critical points, we can take the derivative of \( y \) and set it to zero. However, this can be complex due to the quotient and the square root. Instead, we can analyze the function's behavior qualitatively or use numerical methods to find specific critical points. ### 4. **Summary** - **Domain**: \( x \in \left[\frac{3}{5}, \infty\right) \) - **Behavior**: - \( y \to 0 \) as \( x \to \frac{3}{5} \) - \( y \to \infty \) as \( x \to \infty \) This function is defined for \( x \geq \frac{3}{5} \) and increases without bound as \( x \) increases. Further analysis, such as finding the derivative, would provide more information about local maxima or minima.