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

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

Correct Answer: 5

Step 1: Identify the quadratic function, which is f(x) = -x^2 + 4x + 1.
Step 2: Recognize that this is a downward-opening parabola because the coefficient of x^2 is negative (-1).
Step 3: Find the x-coordinate of the vertex using the formula x = -b/(2a), where a = -1 and b = 4.
Step 4: Calculate x = -4/(2 * -1) = -4/-2 = 2.
Step 5: Substitute x = 2 back into the function to find the maximum value: f(2) = -2^2 + 4*2 + 1.
Step 6: Calculate f(2) = -4 + 8 + 1 = 5.
Step 7: Conclude that the maximum value of the function is 5.

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