if `x+y=10`, thent he maximum value of xy is

Options

A. 5
B. 20
C. 25
D. none of these

Explanation

To find the maximum value of \( xy \) given the constraint \( x + y = 10 \), we can follow these steps:

### Step 1: Express \( y \) in terms of \( x \)
From the equation \( x + y = 10 \), we can express \( y \) as:
\[ y = 10 - x \]

### Step 2: Substitute \( y \) into the product \( xy \)
Now we can substitute \( y \) into the expression for \( xy \):
\[ xy = x(10 - x) = 10x - x^2 \]

### Step 3: Define the function to maximize
Let \( f(x) = 10x - x^2 \). We need to maximize this function.

### Step 4: Differentiate the function
To find the maximum, we differentiate \( f(x) \) with respect to \( x \):
\[ f'(x) = 10 - 2x \]

### Step 5: Set the derivative equal to zero
To find the critical points, we set the derivative equal to zero:
\[ 10 - 2x = 0 \]
Solving for \( x \):
\[ 2x = 10 \implies x = 5 \]

### Step 6: Verify if it is a maximum
Next, we need to check if this critical point is indeed a maximum by finding the second derivative:
\[ f''(x) = -2 \]
Since \( f''(x) \) is negative, this indicates that \( f(x) \) has a maximum at \( x = 5 \).

### Step 7: Find the corresponding value of \( y \)
Substituting \( x = 5 \) back into the equation for \( y \):
\[ y = 10 - 5 = 5 \]

### Step 8: Calculate the maximum value of \( xy \)
Now, we can calculate the maximum value of \( xy \):
\[ xy = 5 \times 5 = 25 \]

Thus, the maximum value of \( xy \) is \( \boxed{25} \).

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