absolute/global minimum
-at c if f(c) is less 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 minimum of a function is the lowest value of the function over its entire domain. It represents the lowest point on the graph of the function and is often written as f(x) = min, where min is the minimum value of the function. To find the absolute minimum of a function, you need to consider all possible values of the independent variable within the domain and evaluate the function at each of these values. Then, you look for the smallest output value of the function.
For example, consider the function f(x) = x^2 + 2x + 3. The domain of this function is all real numbers because it is a polynomial function. To find the absolute minimum of this function, you can use calculus by taking the derivative of the function, setting it equal to zero, and solving for x. The result is x = -1, which is a critical point of the function. You can then evaluate the function at this point to obtain the absolute minimum value of f(x), which is f(-1) = 2.
Alternatively, you can graph the function to visualize its lowest point. In this example, the graph is a parabola that opens upward, indicating that it has a minimum value. By examining the graph, you can see that the vertex of the parabola is located at (-1,2), which is the absolute minimum point of the function.
More Answers:
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