In this article, we will explore the function f(x) = 2x^2 + 3mx – 2m and demonstrate how to find its minimum value (g) and its maximum value as the parameter m varies. The graph above shows the behavior of the function for different values of m, illustrating how the graph changes with m.
Finding the Minimum Value (g)
The function f(x) = 2x^2 + 3mx – 2m has its minimum value when the first derivative is zero. By solving f'(x) = 0, we can determine the critical point and check its concavity with the second derivative. The minimum occurs at x = -3m/4.
Maximum Value of g
To find the maximum value of g as m changes, you can evaluate the minimum values for different values of m and look for the maximum among them. This provides insight into how the function behaves under various conditions.
Conclusion
By understanding the changes in the function’s behavior based on the parameter m, we can make informed conclusions about how to approach problems involving quadratic functions and optimization.
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