Which one of the following is true about A.M. , G.M. and H.M. :

Options

A. H.M. `ge ` G.M. `ge` A.M.
B. A.M. `ge` H.M. `ge` G.M.
C. A.M. `gt` G.M. `gt` H.M.
D. G.M. `gt` H.M. `gt` A.M.

Answer

To determine the relationship between the Arithmetic Mean (A.M.), Geometric Mean (G.M.), and Harmonic Mean (H.M.), we can follow these steps:

### Step 1: Define the Means
- **Arithmetic Mean (A.M.)**: For two numbers \( a \) and \( b \), the A.M. is given by:
\[ A.M. = \frac{a + b}{2} \]

- **Geometric Mean (G.M.)**: For two numbers \( a \) and \( b \), the G.M. is given by:
\[ G.M. = \sqrt{ab} \]

- **Harmonic Mean (H.M.)**: For two numbers \( a \) and \( b \), the H.M. is given by:
\[ H.M. = \frac{2ab}{a + b} \]

### Step 2: Establish the Inequality
We need to establish the relationship among A.M., G.M., and H.M. for any two positive numbers \( a \) and \( b \).

The well-known inequality states:
\[ A.M. \geq G.M. \geq H.M. \]

### Step 3: Prove the Inequality
1. **A.M. vs G.M.**:
- Using the Cauchy-Schwarz inequality, we can show that:
\[ (a + b)^2 \geq 4ab \implies A.M. \geq G.M. \]

2. **G.M. vs H.M.**:
- We can also show that:
\[ \frac{1}{H.M.} = \frac{a + b}{2ab} \leq \frac{1}{G.M.} = \frac{2}{\sqrt{ab}} \implies G.M. \geq H.M. \]

### Step 4: Conclusion
From the established inequalities, we conclude:
\[ A.M. \geq G.M. \geq H.M. \]
Thus, the correct statement about A.M., G.M., and H.M. is that the Arithmetic Mean is the largest, followed by the Geometric Mean, and the Harmonic Mean is the smallest.

### Final Answer
The correct option is:
**C. A.M. ≥ G.M. ≥ H.M.**

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