minimum value of a function
When a parabola opens upward, the y-value of the vertex.
The minimum value of a function is the smallest output (or y-value) that the function can produce. To find the minimum value of a function, we typically use calculus by taking the derivative of the function and setting it to zero (i.e. finding the critical points). We then evaluate the function at each critical point, as well as at the endpoints of the domain if they exist, to determine the minimum value.
For example, consider the function f(x) = x^2 – 4x + 5. To find the minimum value of this function, we take its derivative: f'(x) = 2x – 4. Setting this equal to zero, we get 2x – 4 = 0, or x = 2. This critical point splits the domain of the function into two intervals: (-infinity, 2) and (2, infinity). We evaluate the function at the critical point and the endpoints of the domain: f(2) = 1, f(-infinity) = infinity, and f(infinity) = infinity. Therefore, the minimum value of f(x) is 1, which occurs at x = 2.
It’s important to note that not all functions have a minimum value. For example, the function f(x) = x^2 does not have a minimum value, since it can produce arbitrarily small negative values as x approaches negative infinity.
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