`y=x(c-x)` where `c` is a constant. Find maximum value of `y`.

Explanation

To find the maximum value of the function \( y = x(c - x) \), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ y = x(c - x) \]
This can be expanded to:
\[ y = cx - x^2 \]

### Step 2: Differentiate the function
To find the maximum value, we need to find the critical points by differentiating \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(cx - x^2) = c - 2x \]

### Step 3: Set the derivative to zero
To find the critical points, set the derivative equal to zero:
\[ c - 2x = 0 \]
Solving for \( x \):
\[ 2x = c \implies x = \frac{c}{2} \]

### Step 4: Determine if it's a maximum
To confirm that this critical point is a maximum, we need to check the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(c - 2x) = -2 \]
Since \( \frac{d^2y}{dx^2} < 0 \), this indicates that the function has a maximum at \( x = \frac{c}{2} \).

### Step 5: Find the maximum value of \( y \)
Now substitute \( x = \frac{c}{2} \) back into the original equation to find the maximum value of \( y \):
\[ y_{\text{max}} = \left(\frac{c}{2}\right)\left(c - \frac{c}{2}\right) = \left(\frac{c}{2}\right)\left(\frac{c}{2}\right) = \frac{c^2}{4} \]

Thus, the maximum value of \( y \) is:
\[ \boxed{\frac{c^2}{4}} \]