Question

The arithmetic mean of any two nonnegative real numbers a and b is greater than or equal to their geometric mean vab. [Hint: consider (Va - vb) 0.]

Answer 1

Answer with explanation:

Here, a and b are two real numbers.

Arithmetic Mean of a and b

`A=(a+b)/(2)`

`\rightarrow a< (a+b)/(2)<b`

Geometric Mean of a and b

`G=√(ab)`

`\rightarrow a< √(ab)<b`

`A-G\\\\=(a+b)/(2)-√(ab)\\\\=(a+b-2√(ab))/(2)\\\\=[(√(a)-√(b))/(√(2))]^2>0\\\\A-G>0\\\\A>G`

Square of difference of any two numbers is greater than or equal to 0.

⇒A.M of two Numbers > G.M of two Numbers