To find the maximum value of the function (f(x) = 2 + x - x^2), we can use calculus.

First, let's find the derivative of the function with respect to (x):

(f'(x) = \\frac{d}{dx}(2 + x - x^2))

Applying the power rule of differentiation, we get:

(f'(x) = 0 + 1 - 2x)

(f'(x) = 1 - 2x)

To find the maximum or minimum points of the function, we set the derivative equal to zero and solve for (x):

(1 - 2x = 0)

(2x = 1)

(x = \\frac{1}{2})

Now, to determine whether this point is a maximum or minimum, we can take the second derivative:

(f''(x) = \\frac{d}{dx}(1 - 2x))

(f''(x) = -2)

Since the second derivative is negative, the function has a maximum at (x = \\frac{1}{2}).

Next, let's substitute (x = \\frac{1}{2}) back into the original function to find the corresponding maximum value:

(f\\left(\\frac{1}{2}\\right) = 2 + \\frac{1}{2} - \\left(\\frac{1}{2}\\right)^2)

(f\\left(\\frac{1}{2}\\right) = 2 + \\frac{1}{2} - \\frac{1}{4})

(f\\left(\\frac{1}{2}\\right) = \\frac{9}{4})

Therefore, the maximum value of the function (f(x) = 2 + x - x^2) is ( \\frac{9}{4} ).