If performance of the students 'y' depends on the number of hours 'x' given by the relation :
`y=4x-(x^(2))/(2)`.
find the number of hours, the students work to have the best performance.

Answer

To solve the problem of finding the number of hours \( x \) that students should work to achieve the best performance \( y \), we will follow these steps:

### Step 1: Write down the given function
The performance \( y \) is given by the equation:
\[ y = 4x - \frac{x^2}{2} \]

### Step 2: Differentiate the function
To find the maximum performance, we need to differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}\left(4x - \frac{x^2}{2}\right) \]
Calculating the derivative:
\[ \frac{dy}{dx} = 4 - \frac{1}{2}(2x) = 4 - x \]

### Step 3: Set the derivative to zero
To find the critical points, we set the derivative equal to zero:
\[ 4 - x = 0 \]
Solving for \( x \):
\[ x = 4 \]

### Step 4: Verify if it is a maximum using the second derivative
Next, we need to check whether this critical point is a maximum by finding the second derivative:
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(4 - x) = -1 \]
Since the second derivative is negative (\(-1 < 0\)), this indicates that the function \( y \) has a maximum at \( x = 4 \).

### Conclusion
Thus, the number of hours the students should work to achieve the best performance is:
\[ \boxed{4} \]