What is the maximum value of f(x) = -x^2 + 4x + 1? (2023)

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Question: What is the maximum value of f(x) = -x^2 + 4x + 1? (2023)

Options:

Correct Answer: 5

Solution:

The vertex occurs at x = 2. f(2) = -2^2 + 4(2) + 1 = 5.

What is the maximum value of f(x) = -x^2 + 4x + 1? (2023)

What is the maximum value of f(x) = -x^2 + 4x + 1? (2023)

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 f(x) = ax^2 + bx + c, where a = -1, b = 4, and c = 1.
Step 3: Find the x-coordinate of the vertex using the formula x = -b/(2a). Here, b = 4 and a = -1.
Step 4: Calculate x = -4/(2 * -1) = -4 / -2 = 2.
Step 5: Now that we have x = 2, we need to find the maximum value of the function by substituting x back into f(x).
Step 6: Calculate f(2) = -2^2 + 4(2) + 1.
Step 7: Simplify f(2) = -4 + 8 + 1 = 5.
Step 8: Therefore, the maximum value of f(x) is 5.

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