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

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

Options:

Correct Answer: x = 2

Solution:

The vertex of the parabola given by f(x) = -x^2 + 4x + 1 occurs at x = -b/(2a) = -4/(-2) = 2, which gives the maximum value.

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

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

Step 1: Identify the function you are working with, which is f(x) = -x^2 + 4x + 1.
Step 2: Recognize that this function is a quadratic function, which forms a parabola.
Step 3: Note that the coefficient of x^2 is negative (-1), meaning the parabola opens downwards and has a maximum point.
Step 4: To find the x-coordinate of the vertex (maximum point), use the formula x = -b/(2a).
Step 5: Identify the values of a and b from the function: a = -1 and b = 4.
Step 6: Substitute the values of a and b into the formula: x = -4/(-2).
Step 7: Calculate -4/(-2) which equals 2. This means the maximum value occurs at x = 2.
Step 8: To find the maximum value of the function, substitute x = 2 back into the original function f(x).
Step 9: Calculate f(2) = -2^2 + 4(2) + 1 = -4 + 8 + 1 = 5.
Step 10: Conclude that the maximum value of the function occurs at x = 2 and the maximum value is 5.

Quadratic Functions – Understanding the properties of quadratic functions, including their vertex and maximum/minimum values.
Vertex Formula – Using the vertex formula x = -b/(2a) to find the x-coordinate of the vertex of a parabola.
Maxima and Minima – Identifying whether a quadratic function opens upwards or downwards to determine if the vertex is a maximum or minimum.