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

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

Step 1: Identify the function we are working with, which is f(x) = -x^2 + 4x + 1.
Step 2: To find the maximum value, we need to calculate the derivative of the function, denoted as f'(x).
Step 3: The derivative f'(x) is calculated as f'(x) = -2x + 4.
Step 4: Set the derivative equal to zero to find critical points: -2x + 4 = 0.
Step 5: Solve for x by adding 2x to both sides: 4 = 2x.
Step 6: Divide both sides by 2 to find x: x = 2.
Step 7: To confirm that this point is a maximum, we calculate the second derivative, f''(x).
Step 8: The second derivative f''(x) is f''(x) = -2.
Step 9: Since f''(x) = -2 is less than 0, this indicates that the function has a maximum at x = 2.

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