If p and q are positive real numbers such that `p^2+q^2=1` , then the maximum value of p+q is

Options

A. `1/sqrt2`
B. `sqrt2`
C. 2
D. `1/2`

Explanation

To find the maximum value of \( p + q \) given that \( p^2 + q^2 = 1 \) and both \( p \) and \( q \) are positive real numbers, we can use the method of Lagrange multipliers or apply the Cauchy-Schwarz inequality. Here, we'll use the Cauchy-Schwarz inequality for simplicity.

### Step-by-step Solution:

1. **Understanding the Condition**: We know that \( p^2 + q^2 = 1 \). This represents a quarter of a circle in the first quadrant of the Cartesian plane.

2. **Applying Cauchy-Schwarz Inequality**: According to the Cauchy-Schwarz inequality, we have:
\[ (p + q)^2 \leq (1^2 + 1^2)(p^2 + q^2) \]
Here, we set \( a_1 = p \), \( a_2 = q \), and \( b_1 = 1 \), \( b_2 = 1 \).

3. **Substituting the Values**: Since \( p^2 + q^2 = 1 \), we substitute:
\[ (p + q)^2 \leq (1 + 1)(1) = 2 \]

4. **Taking the Square Root**: Taking the square root of both sides gives:
\[ p + q \leq \sqrt{2} \]

5. **Finding When Equality Holds**: The equality in Cauchy-Schwarz holds when:
\[ \frac{p}{1} = \frac{q}{1} \implies p = q \]
This means that \( p \) and \( q \) are equal.

6. **Substituting Back to Find Values**: If \( p = q \), then from the condition \( p^2 + q^2 = 1 \):
\[ 2p^2 = 1 \implies p^2 = \frac{1}{2} \implies p = \frac{1}{\sqrt{2}} \quad \text{and} \quad q = \frac{1}{\sqrt{2}} \]

7. **Calculating Maximum Value**: Therefore, the maximum value of \( p + q \) is:
\[ p + q = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2} \]

### Final Answer:
The maximum value of \( p + q \) is \( \sqrt{2} \).

---