For the function f(x) = -x^2 + 4x + 1, find the maximum value. (2023)
Practice Questions
Q1
For the function f(x) = -x^2 + 4x + 1, find the maximum value. (2023)
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Questions & Step-by-Step Solutions
For the function f(x) = -x^2 + 4x + 1, find the maximum value. (2023)
Step 1: Identify the function f(x) = -x^2 + 4x + 1.
Step 2: Recognize that this is a quadratic function in the form f(x) = ax^2 + bx + c, where a = -1, b = 4, and c = 1.
Step 3: To find the maximum value, use the formula for the x-coordinate of the vertex: x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -4/(-2).
Step 5: Calculate -4/(-2) to find x = 2.
Step 6: Now, substitute x = 2 back into the function to find the maximum value: f(2) = -2^2 + 4(2) + 1.
Step 7: Calculate f(2): f(2) = -4 + 8 + 1 = 5.
Step 8: The maximum value of the function is 5.
Quadratic Functions – Understanding the properties of quadratic functions, including how to find their maximum or minimum values using the vertex formula.
Vertex Formula – Using the vertex formula x = -b/(2a) to find the x-coordinate of the vertex, which gives the maximum or minimum value of the quadratic function.
Function Evaluation – Evaluating the function at the vertex to find the maximum or minimum value.
