If a, b, c are non-negative real numbers such that , then prove that the product abc cannot exceed 1.
Given that , , , so certainly , , , and .
Now, and hence, , hence we get:
. Clearly, the presence of and reminds us of the AM-GM inequality.
Here it is .
Also, we can say: . Now, let .
that is, , or , that is, . So, this is a beautiful application of arithmetic mean-geometric mean inequality twice. 🙂 🙂
If a, b, c are three rational numbers, then prove that : is always the square of a rational number.
Let , , . It can be very easily shown that , or . So, the given expression is a perfect square !!! BINGO! 🙂 🙂 🙂