Relative Minumum
A relative minimum is a concept in calculus that refers to the lowest point of a function within a specific interval
A relative minimum is a concept in calculus that refers to the lowest point of a function within a specific interval. More specifically, a point (x, f(x)) on a graph is considered a relative minimum if it is the lowest point in a small neighborhood around that point.
To determine if a point is a relative minimum, we need to examine the behavior of the function in the surrounding area. If the function is decreasing as we move to the left or right of the point, then that point can be considered a relative minimum.
Mathematically, for a function f(x) to have a relative minimum at x = c, three conditions must be met:
1. The function must be defined at x = c.
2. The derivative of the function, f'(x), must exist and be equal to zero at x = c.
3. The second derivative of the function, f”(x), must exist and be positive at x = c.
Let’s illustrate this with an example: Consider the function f(x) = x^2 – 4x + 3. To find the relative minimum, we can follow these steps:
1. Determine the derivative of the function: f'(x) = 2x – 4.
2. Find the critical points by setting f'(x) = 0 and solve for x: 2x – 4 = 0. Solving this equation gives x = 2.
3. Calculate the second derivative of the function: f”(x) = 2.
4. Evaluate the second derivative at the critical point x = 2. Since f”(2) = 2 > 0, this condition is satisfied.
5. Finally, evaluate f(x) at x = 2 to determine the y-coordinate of the relative minimum: f(2) = 2^2 – 4(2) + 3 = -1.
Therefore, the function f(x) = x^2 – 4x + 3 has a relative minimum at the point (2, -1), where x = 2 is the x-coordinate and -1 is the y-coordinate.
It is worth noting that a function can have multiple relative minima within a given interval, or it might not have any at all. It depends on the behavior of the function and the characteristics of its graph.
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