absolute/global maximum
-at c if f(c) is greater than or equal to f(x) for all x in the domain of f-on a given interval, it may be the endpoints (which may or may not be a critical point)-there can be more than one if they have the same y value
The absolute or global maximum of a function represents the highest value the function attains over its entire domain. To find the absolute maxima of a function, you need to check its values at every critical point and at the end-points of its domain.
Let’s consider the function f(x) = x^3 – 3x^2 + 2x on the interval [-1, 3]. To find the absolute maxima of this function, we need to follow these steps:
1. Find the critical points by taking the derivative of the function and setting it equal to zero: f'(x) = 3x^2 – 6x + 2. Solving f'(x) = 0 gives x = 1/3, 2.
2. Evaluate the function at each critical point and at the end-points of the interval: f(-1) = 6, f(1/3) = 25/27, f(2) = 0, f(3) = 8.
3. Compare the values to find the largest one: the absolute maximum of f(x) on [-1, 3] is 25/27, which occurs at x = 1/3.
Therefore, the absolute maximum of f(x) is 25/27, and it occurs at x = 1/3.
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