For any real number x there is a unique integer n such that n≤x<n+1, and the greatest integer function is defined as ⌊x⌋=n. Where are the critical values of the greatest integer function? Which are local maxima and which are local minima?
⌊x⌋ has no maxima and minima. It is a piecewise linear function, and each piece is a horizontal segment of length 1.
Since a critical point is where the derivative is zero or undefined, then that would be every real number, since
at the integers, ⌊x⌋ is discontinuous, so the derivative is undefined.
elsewhere, the derivative is zero.
The critical values of a function are the values of x for which the derivative is either zero or undefined. However, for the greatest integer function, it is not differentiable at integer values, leading to undefined derivatives. Therefore, we need to examine the critical points by considering the left and right limits.
Since the greatest integer function is only defined for real numbers, the critical values occur when x takes on integer values, but it is necessary to determine whether they are local maxima or local minima. Let's analyze the behavior of the greatest integer function around these critical points.
For any given integer value n, we examine the left-hand limit and the right-hand limit as x approaches n from the left and right, respectively.
1. Left-hand limit (n-):
For x < n, the greatest integer function would be n-1. Hence, as x approaches n from the left, the function does not change its value until x reaches n-1, at which point it jumps to n. Therefore, the left-hand limit is n-1.
2. Right-hand limit (n+):
For x > n, the greatest integer function would be n. As x approaches n from the right, the function remains constant at n, without changing its value. Therefore, the right-hand limit is n.
By comparing the left-hand limit, right-hand limit, and the function value at the critical point, we can identify whether it is a local maximum or a local minimum.
Critical points:
The critical values of the greatest integer function are the integers themselves, n.
Local Maxima or Minima:
1. Local maximum:
If the left-hand limit is greater than the function value and greater than the right-hand limit, it is a local maximum. In other words, if n-1 > n and n-1 > n, it is a local maximum.
2. Local minimum:
If the right-hand limit is greater than the function value and greater than the left-hand limit, it is a local minimum. In other words, if n > n and n > n-1, it is a local minimum.
Consequently, for the greatest integer function, the critical points (integer values) are not local maximum or minimum since their left-hand limits and right-hand limits are always equal.
In summary, the critical values of the greatest integer function are the integers, but none of them are local maxima or local minima due to the nature of the function.