The maximum value of the function f(x) = -x^2 + 4x + 1 is:

The maximum value of the function f(x) = -x^2 + 4x + 1 is:

Step 1: Identify the function we are working with, which is f(x) = -x^2 + 4x + 1.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = -1, b = 4, and c = 1.
Step 3: To find the maximum value, we need to find the x-coordinate of the vertex of the parabola. Use the formula x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -4/(2 * -1).
Step 5: Calculate the value: x = -4 / -2 = 2.
Step 6: Now, we need to find the maximum value of the function by evaluating f(2).
Step 7: Substitute x = 2 into the function: f(2) = -2^2 + 4*2 + 1.
Step 8: Calculate f(2): f(2) = -4 + 8 + 1 = 5.
Step 9: Therefore, the maximum value of the function f(x) is 5.

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