+0

# help

0
115
1

Find the minimum value of a/b + b/a, where a and b are positive real numbers.

Dec 4, 2019

#1
+24388
+3

Find the minimum value of $$\dfrac{a}{b} + \dfrac{b}{a}$$, where a and b are positive real numbers.

$$\mathbf{\huge{AM \geq GM }}$$

$$\begin{array}{|rcll|} \hline \dfrac{a^2+b^2}{2} &\geq& \sqrt{a^2b^2} = ab \\ \dfrac{a^2+b^2}{2} &\geq& ab \quad | \quad * \dfrac{2}{ab} \\ \dfrac{a^2+b^2}{ab} &\geq& 2 \\ \dfrac{a^2}{ab}+\dfrac{b^2}{ab} &\geq& 2 \\ \mathbf{\dfrac{a}{b}+\dfrac{b}{a}} &\geq& \mathbf{2} \\ \hline \end{array}$$

The minimum value of $$\dfrac{a}{b} + \dfrac{b}{a}$$ is 2

Dec 4, 2019