Find the maximum value of f(x) = -2x^2 + 10x - 12. (2023)
Practice Questions
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Find the maximum value of f(x) = -2x^2 + 10x - 12. (2023)
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Questions & Step-by-Step Solutions
Find the maximum value of f(x) = -2x^2 + 10x - 12. (2023)
Step 1: Identify the function we need to analyze, which is f(x) = -2x^2 + 10x - 12.
Step 2: Recognize that this is a quadratic function in the form f(x) = ax^2 + bx + c, where a = -2, b = 10, and c = -12.
Step 3: To find the maximum value, we need to find the x-coordinate of the vertex of the parabola. The formula for the x-coordinate of the vertex is x = -b/(2a).
Step 4: Substitute the values of a and b into the formula: x = -10/(2 * -2).
Step 5: Calculate the denominator: 2 * -2 = -4, so x = -10 / -4 = 2.5.
Step 6: Now, we need to find the maximum value of the function at x = 2.5. Substitute x = 2.5 into the function: f(2.5) = -2(2.5^2) + 10(2.5) - 12.
Step 7: Calculate 2.5^2, which is 6.25. Then, multiply by -2: -2 * 6.25 = -12.5.
Step 8: Calculate 10 * 2.5, which is 25.
Step 9: Now combine the results: f(2.5) = -12.5 + 25 - 12.
Step 10: Simplify: -12.5 + 25 = 12.5, and then 12.5 - 12 = 0.5.
Step 11: Therefore, the maximum value of f(x) is 0.5.
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 of a parabola.
Function Evaluation – Evaluating the function at the vertex to find the maximum or minimum value.
