Finding arithmetic mean using geometric mean information

Question

The product of two integers is $10$

Compare :

Quantity A : The arithmetic mean of the integers
Quantity B : $3$

Options:

A) Quantity A > Quantity B

B) Quantity A < Quantity B

C) Quantity A = Quantity B

D) Cannot be determined

---

My solution:

Ive compared it using the relation AM >= GM Suppose the numbers be a and b

AM >= GM

$\frac{a+b}{2} \ge \sqrt{ab}$

$\frac{a+b}{2} \ge \sqrt{10}$

$\frac{a+b}{2} \ge 3.162$

Hence, Option A) Quantity A > Quantity B

---

Is this solution correct as the answer to this question is given as option D) Cannot be determined

Answer 1

hint

Suppose $x>0$ and compare $$f (x)=x+\frac {10}{x} $$ with $6$.

$$f'(x)=\frac {x^2-10}{x^2}$$

the minimum is $f(\sqrt {10})=2\sqrt {10}>6$ hence $A>B $.

If $x <0$ then $f (x)<0 <6$.