To determine whether the Mean Value Theorem (MVT) applies to the function $y = \ln(2x + 3)$ over the interval $\left[-\frac{3}{2}, 0\right]$, we need to check the hypotheses of the MVT:

Hypotheses of the Mean Value Theorem:

1. Continuity on the closed interval $[a, b]$: The function must be continuous on the interval $\left[-\frac{3}{2}, 0\right]$.
2. Differentiability on the open interval $(a, b)$: The function must be differentiable on the interval $\left(-\frac{3}{2}, 0\right)$.

Checking the Function $y = \ln(2x + 3)$:

1. Continuity:

   • The function $y = \ln(2x + 3)$ is defined wherever $2x + 3 > 0$.
   • Solving $2x + 3 > 0$ gives $x > -\frac{3}{2}$, so the function is continuous on the interval $\left[-\frac{3}{2}, 0\right]$.

2. Differentiability:

   • The function $y = \ln(2x + 3)$ is differentiable wherever it is defined. Since $y$ is continuous on the interval $\left[-\frac{3}{2}, 0\right]$ and differentiable on the open interval $\left(-\frac{3}{2}, 0\right)$, it satisfies the differentiability condition.

Conclusion:

Both conditions of the Mean Value Theorem are satisfied:

• $y = \ln(2x + 3)$ is continuous on $\left[-\frac{3}{2}, 0\right]$.
• $y = \ln(2x + 3)$ is differentiable on $\left(-\frac{3}{2}, 0\right)$.

Therefore, the Mean Value Theorem applies to $y = \ln(2x + 3)$ over the interval $\left[-\frac{3}{2}, 0\right]$. This means there exists a $c \in \left(-\frac{3}{2}, 0\right)$ such that:

$y'(c) = \frac{y(0) - y\left(-\frac{3}{2}\right)}{0 + \frac{3}{2}}$

Do you want more details or have any questions?

Related Questions:

1. What is the Mean Value Theorem, and how is it generally used in calculus?
2. Can you explain how to find the derivative of $y = \ln(2x + 3)$?
3. What happens if one of the conditions for the Mean Value Theorem is not satisfied?
4. How do we determine the exact value of $c$ that satisfies the MVT in this case?
5. What is the importance of continuity and differentiability in calculus?

Tip:

When verifying the conditions for the Mean Value Theorem, always check where the function is undefined or discontinuous to ensure accurate conclusions.