Minimum of f(x)

Lori Pearman
EMT 725

Problem: Use the Arithmetic Mean -- Geometric Mean Inequality to show that
Min f(x) = 1/x + x is 2.

Solution:
The inequality is true for non-negative varaibles. So for this problem, we can find the minimum value of f(x) for positive values of x. The inequality for this problem is as follows:
(1/x + x)/2 > or = [(1/x ) x]^(1/2)
1/x + x > or = 2(1) = 2
So f(x) > or = 2.

This implies that for positive values of x, the minimum value of f(x) is 2.
This occurs when 1/x = x. (Or when x = 1.)

The below graph verifies that for positive x values, the minimum point is (1,2).

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