Find the maximum value of f(x) = -x^2 + 4x.

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Question: Find the maximum value of f(x) = -x^2 + 4x.

Options:

Correct Answer: 4

Solution:

The vertex form gives maximum at x = 2. f(2) = -2^2 + 4*2 = 4.

Find the maximum value of f(x) = -x^2 + 4x.

Find the maximum value of f(x) = -x^2 + 4x.

Correct Answer: 4

Step 1: Identify the function you need to analyze, which is f(x) = -x^2 + 4x.
Step 2: Recognize that this is a quadratic function in the form of ax^2 + bx + c, where a = -1, b = 4, and c = 0.
Step 3: Since the coefficient of x^2 (a) is negative, the parabola opens downwards, meaning it has a maximum point.
Step 4: To find the x-coordinate of the vertex (maximum point), use the formula x = -b/(2a). Here, b = 4 and a = -1.
Step 5: Calculate x = -4/(2 * -1) = -4/-2 = 2.
Step 6: Now, substitute x = 2 back into the function to find the maximum value: f(2) = -2^2 + 4*2.
Step 7: Calculate f(2) = -4 + 8 = 4.
Step 8: Therefore, the maximum value of f(x) is 4.

Quadratic Functions – Understanding the properties of quadratic functions, including their vertex and maximum/minimum values.
Vertex Form – Using the vertex form of a quadratic equation to find the maximum or minimum point.