I think you mean ##x^2 +x + 1\leq M##? Otherwise it would be a bit pointless.I've assumed there is a M<3/4 that x^2 +x + 1>M
An infimum is the greatest lower bound of a set of numbers. In other words, it is the smallest number in the set that is still greater than or equal to all other numbers in the set.
To prove that 3/4 is the infimum of A, we need to show that it is the smallest number that is greater than or equal to all elements in the set A. This can be done by showing that 3/4 is less than or equal to each element in A, and that there is no smaller number that satisfies this condition.
Finding the infimum of a set is important because it can provide valuable information about the set, such as its range and bounds. It also helps in understanding the behavior of the set and can be useful in solving optimization problems.
The set A = { x^2 +x + 1 } is the set of all possible values that can be obtained by plugging in different values for x in the given equation. It is important in this question because we are trying to find the smallest number in this set that is greater than or equal to all other numbers in the set.
Yes, for example, to prove that 3/4 is the infimum of A = { x^2 +x + 1 }, we can show that it is less than or equal to each element in the set. For instance, when x = 0, we get 3/4 = 0^2 + 0 + 1 = 1, which is less than or equal to 1. Similarly, when x = 1/2, we get 3/4 = (1/2)^2 + 1/2 + 1 = 3/4, which is less than or equal to 3/4. And there is no smaller number that satisfies this condition, thus proving that 3/4 is the infimum of A.