https://file.aiquickdraw.com/mathbot/file/17282913598199vg2m990.jpg

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.

Using Lagrange’s mean value theorem, prove that \( \frac{b - a}{b} < \ln\left(\frac{b}{a}\right) < \frac{b - a}{a} \) for \( 0 < a < b \). Hence show that \( \frac{1}{4} < \ln\left(\frac{4}{3}\right) < \frac{1}{3} \).

Learn how to prove the inequality \( \frac{b - a}{b} < \ln\left(\frac{b}{a}\right) < \frac{b - a}{a} \) using Lagrange’s Mean Value Theorem for \( 0 < a < b \). Understand the application of the theorem in logarithmic functions and inequalities, with a step-by-step breakdown.

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