What is the smallest value of positive constant that will make greater than or equal to zero for all positive values of ?
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This is a clear aplication of the Arithmetic-Geometric Inequality.
Let rewrite the sentence as − 1 + m x + ( x 1 ) .
Applying the Arithmetic-Geometric Inequality, we have:
− 1 + m x + ( x 1 ) ≥ − 1 + 2 m x ( x 1 ) = − 1 + 2 m
Since we want our sentence greater than or equal to zero, just do:
− 1 + 2 m ≥ 0
Therefore, 2 m ≥ 1
Hence we have that m ≥ 4 1 or m ≤ − 4 1 .
But we want the smallest positive integer, then m ≥ 4 1
Thus, we conclude that the smallest integer m such that − 1 + m x + ( x 1 ) is greater than or equal to zero is 4 1 = 0 . 2 5
Q . E . D .