Question

find the minimum value of ax+by where xy=c^2

Answer 1

 the minimum value is this >*c*sq root(ab)

Answer 2

Given:

xy=c^2

To Find:

The minimum value of ax+by

Solution:

For any number of terms  a1, a2, a3 , .. an ,

Arithmetic Mean of the terms is always greater than equal to the geometric mean of the terms.

If we consider a sequence containing two terms ax and by ,

• Arithmetic mean of ax and by = (ax + by)/2

• Geometric mean of ax and by =$\sqrt{ (ax * by)}$

Applying:

• AM ≥ GM

• (ax + by)/2 ≥ $\sqrt{ (ax * by)}$

Given that ,

• xy = c²
• ax+ by ≥ 2$\sqrt{ (ab *xy)}$
• ax + by ≥ 2√ab √c²
• ax + by ≥ 2c√ab

The minimum value of ax+by where xy=c^2 is 2c√ab.