If `ab=25`, then the minimum value of `a+b` is :

Options

A. 10
B. 8
C. 24
D. 16

Answer

To find the minimum value of \( a + b \) given that \( ab = 25 \), we can use the method of substitution and the properties of arithmetic and geometric means.

### Step 1: Set up the equations
We know that:
\[ ab = 25 \]
We want to minimize:
\[ a + b \]

### Step 2: Use the AM-GM inequality
According to the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
This implies:
\[ a + b \geq 2\sqrt{ab} \]

### Step 3: Substitute the value of \( ab \)
Substituting \( ab = 25 \) into the inequality:
\[ a + b \geq 2\sqrt{25} \]
Calculating the right side:
\[ a + b \geq 2 \times 5 = 10 \]

### Step 4: Determine if the minimum value can be achieved
The equality in the AM-GM inequality holds when \( a = b \). Thus, we set:
\[ a = b \]
Substituting into the equation \( ab = 25 \):
\[ a^2 = 25 \implies a = 5 \quad \text{and} \quad b = 5 \]

### Step 5: Calculate \( a + b \)
Now we can find:
\[ a + b = 5 + 5 = 10 \]

### Conclusion
Therefore, the minimum value of \( a + b \) is:
\[ \boxed{10} \]