The question in the image asks to:

1. Prove using Lagrange’s Mean Value Theorem (MVT) that: $\frac{b - a}{b} < \ln\left(\frac{b}{a}\right) < \frac{b - a}{a}$ for $0 < a < b$.

2. Hence show that: $\frac{1}{4} < \ln\left(\frac{4}{3}\right) < \frac{1}{3}$

Step 1: Applying Lagrange’s Mean Value Theorem (MVT)

Lagrange's MVT states that for a differentiable function $f(x)$ on the interval $[a, b]$, there exists a point $c \in (a, b)$ such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$

Let’s apply this theorem to the function $f(x) = \ln(x)$, which is continuous and differentiable on $(0, \infty)$.

For the interval $[a, b]$ (where $0 < a < b$), we have: $\ln(b) - \ln(a) = f'(c)(b - a)$ This simplifies to: $\ln\left(\frac{b}{a}\right) = \frac{1}{c}(b - a)$ where $c \in (a, b)$.

Since $c \in (a, b)$, we know that $a < c < b$. This gives the bounds: $\frac{1}{b} < \frac{1}{c} < \frac{1}{a}$ Multiplying this inequality by $(b - a)$, we get: $\frac{b - a}{b} < \ln\left(\frac{b}{a}\right) < \frac{b - a}{a}$

Thus, the first part of the inequality is proven.

Step 2: Proving the specific case $\frac{1}{4} < \ln\left(\frac{4}{3}\right) < \frac{1}{3}$

Let’s apply the proven inequality for $a = 3$ and $b = 4$.

• First, compute $\frac{b - a}{b} = \frac{4 - 3}{4} = \frac{1}{4}$.
• Then, compute $\frac{b - a}{a} = \frac{4 - 3}{3} = \frac{1}{3}$.

So, we now have: $\frac{1}{4} < \ln\left(\frac{4}{3}\right) < \frac{1}{3}$

This completes the proof for the second part.

Would you like further explanation or details on any step?

5 Related Questions:

1. What is the geometric interpretation of Lagrange’s Mean Value Theorem?
2. How can you derive tighter bounds for $\ln\left(\frac{b}{a}\right)$?
3. Why is the function $\ln(x)$ used in this type of inequality proof?
4. Can you apply the same theorem to functions other than logarithmic ones?
5. What happens to the inequality as $a$ approaches $b$?

Tip:

When using the Mean Value Theorem, it's helpful to always identify where the derivative function’s behavior influences the bounds of the inequality.