cauchy

The Cauchy Schwarz inequality is a relatively rare concept that is unknown to quite a few aspirants. This does not appear commonly in entrance tests but the results can be used in certain questions which deal with inequalities.

If the question involves sum of terms and product of these terms in some order, we use the Arithmetic mean > Geometric mean principle. However, if the question involves sum of squares, you would probably need to take the help of the Cauchy Schwarz inequality.

The concept simply says that:

cauchy1

It is pretty easy to understand the concept but solving questions might be a bit difficult especially if you are not able to visualize the pairs that you need to take. Let’s see this concept with the help of a few examples.

What will be the minimum value of the following expression?

cauchy2

(a) 2             (b) 4             (c) 5             (d) 6

cauchy3

So, the minimum value will be 6.

Short theorem on Cauchy Schwarz inequality

cauchy4

Where, a1, a2, a3… are real numbers and x1, x2, x3… are positive real numbers.

If a + b + c = 7 what is the minimum value of a2 + 9b2 + 4c2?

Here, it is obvious that a, 3b and 2c will be in the LHS. Now, we already have the value of a + b + c = 7 and so, we need to make sure that the other part in the LHS cancels out the coefficients of a2, 9b2 and 4c2 so that multiplying the two gives us the form of (a + b + c)2 on the RHS.

cauchy5

So, the minimum value of the expression is 36.

What is the minimum value of the expression?

cauchy6

The level of difficulty is pretty high for these questions involving the Cauchy Schwarz inequality and unless you have understood the concept fully, it does not make any sense to jump on to such questions at the start. If you are fairly confident about the method it would be a huge differentiator when it comes to the final tally of scores. When it comes to entrance tests, probably something similar to the a2, 9b2 and 4c2 question can be expected. The rest are way above CAT level and are meant to be understood as concepts only.

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