If a and b are positive real numbers such that `a+b =c,` then the minimum value of `((4 )/(a)+ (1)/(b))` is equal to :

Options

A. `2/3`
B. `1/3`
C. `1`
D. `3/2`

Answer

To find the minimum value of the expression \( \frac{4}{a} + \frac{1}{b} \) given that \( a + b = c \), we can use the method of Lagrange multipliers or apply the Cauchy-Schwarz inequality. Here, we will use the Cauchy-Schwarz inequality for simplicity.

### Step-by-step Solution:

1. **Given Information**:
We have \( a + b = c \) where \( a \) and \( b \) are positive real numbers.

2. **Rearranging the Expression**:
We need to minimize \( \frac{4}{a} + \frac{1}{b} \).

3. **Applying Cauchy-Schwarz Inequality**:
According to the Cauchy-Schwarz inequality:
\[ (x_1^2 + x_2^2)(y_1^2 + y_2^2) \geq (x_1y_1 + x_2y_2)^2 \]
We can set \( x_1 = \frac{2}{\sqrt{a}} \), \( x_2 = \frac{1}{\sqrt{b}} \), \( y_1 = \sqrt{a} \), and \( y_2 = \sqrt{b} \).

4. **Setting Up the Inequality**:
Applying Cauchy-Schwarz:
\[ \left(\frac{2}{\sqrt{a}} + \frac{1}{\sqrt{b}}\right)^2 \leq \left(2^2 + 1^2\right)\left(\frac{1}{a} + \frac{1}{b}\right) \]
This simplifies to:
\[ \left(\frac{4}{a} + \frac{1}{b}\right) \geq \frac{(2 + 1)^2}{1} = 9 \]

5. **Using the Constraint**:
Since \( a + b = c \), we can express \( b \) in terms of \( a \):
\[ b = c - a \]
Substitute this into our inequality:
\[ \frac{4}{a} + \frac{1}{c - a} \geq 9 \]

6. **Finding the Minimum Value**:
To find the minimum value, we can differentiate the expression \( \frac{4}{a} + \frac{1}{c - a} \) with respect to \( a \) and set the derivative to zero. However, we can also find the minimum value by testing specific values of \( a \) and \( b \) that satisfy the constraint.

7. **Testing Values**:
Let \( a = 4 \) and \( b = 1 \) (since \( a + b = 5 \)):
\[ \frac{4}{4} + \frac{1}{1} = 1 + 1 = 2 \]
This shows that the minimum value is indeed \( 2 \).

### Conclusion:
The minimum value of \( \frac{4}{a} + \frac{1}{b} \) when \( a + b = c \) is \( 2 \).