If A be any event in sample space then the maximum value of `3sqrt(P(A))+4sqrt(P(barA))` is :

Options

A. 4
B. 2
C. 5
D. Can not determined

Answer

To find the maximum value of the expression \(3\sqrt{P(A)} + 4\sqrt{P(\bar{A})}\), we will follow these steps:

### Step 1: Define the probabilities
Let \(P(A) = x\). Then, since \(P(\bar{A}) = 1 - P(A)\), we have:
\[ P(\bar{A}) = 1 - x \]

### Step 2: Rewrite the expression
We can rewrite the expression in terms of \(x\):
\[ f(x) = 3\sqrt{x} + 4\sqrt{1 - x} \]

### Step 3: Find the derivative
To find the maximum value of \(f(x)\), we need to find its derivative and set it to zero:
\[ f'(x) = \frac{d}{dx}(3\sqrt{x}) + \frac{d}{dx}(4\sqrt{1 - x}) \]
Calculating the derivatives:
\[ f'(x) = 3 \cdot \frac{1}{2\sqrt{x}} - 4 \cdot \frac{1}{2\sqrt{1 - x}} \cdot (-1) \]
This simplifies to:
\[ f'(x) = \frac{3}{2\sqrt{x}} + \frac{4}{2\sqrt{1 - x}} = \frac{3}{2\sqrt{x}} + \frac{2}{\sqrt{1 - x}} \]

### Step 4: Set the derivative to zero
Set \(f'(x) = 0\):
\[ \frac{3}{2\sqrt{x}} = \frac{2}{\sqrt{1 - x}} \]
Cross-multiplying gives:
\[ 3\sqrt{1 - x} = 4\sqrt{x} \]

### Step 5: Square both sides
Squaring both sides results in:
\[ 9(1 - x) = 16x \]
Expanding and rearranging:
\[ 9 - 9x = 16x \implies 9 = 25x \implies x = \frac{9}{25} \]

### Step 6: Find the maximum value
Now substitute \(x = \frac{9}{25}\) back into the expression for \(f(x)\):
\[ f\left(\frac{9}{25}\right) = 3\sqrt{\frac{9}{25}} + 4\sqrt{1 - \frac{9}{25}} \]
Calculating each term:
\[ = 3 \cdot \frac{3}{5} + 4\sqrt{\frac{16}{25}} = \frac{9}{5} + 4 \cdot \frac{4}{5} \]
\[ = \frac{9}{5} + \frac{16}{5} = \frac{25}{5} = 5 \]

### Conclusion
Thus, the maximum value of \(3\sqrt{P(A)} + 4\sqrt{P(\bar{A})}\) is:
\[ \boxed{5} \]