Understanding Local Minima in Mathematics: The Relationship Between f(c) and f(x)

If f (c) ≤ f (x) for every x in the domain of f, then the point (c, f (c)) is a local minimum.

FALSE…The point is an absolute minimum. Local minima must be critical numbers,and this statement says nothing about critical numbers.

Yes, if f(c) ≤ f(x) for every x in the domain of f, then the point (c, f(c)) is a local minimum.

A local minimum is a point on the function where the value of the function is lower than all the neighboring points within a certain interval. Therefore, if f(c) is less than or equal to the value of f(x) for all x in the domain of f, it means that f(c) is the lowest value that f can achieve within that interval. Hence, f(c) is a local minimum.

One way to visualize this is by looking at a graph of the function. If we plot the function on a coordinate system, then the point (c, f(c)) represents a point on the curve. Since f(c) is the lowest point on the curve within a certain interval, it means that the curve slopes upwards on both sides of the point (c, f(c)). This slope ensures that there are no other local minima within that interval.

Overall, this shows that if f(c) ≤ f(x) for every x in the domain of f, then the point (c, f(c)) is a local minimum.

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