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  • If , then the minimum value of <img alt="\frac{ab}{c^{2}}+\frac{bc}{a^{2}}+\frac{ca}{b^{2}}" src="https:/

If a,b,c>0, then the minimum value of \frac{ab}{c^{2}}+\frac{bc}{a^{2}}+\frac{ca}{b^{2}} is

Option: 1

1


Option: 2

2


Option: 3

3


Option: 4

4


Answers (1)

best_answer

Let \;a_{1}=\frac{a b}{c^{2}}, a_{2}=\frac{b c}{a^{2}}, a_{3}=\frac{c a}{b^{2}}

and apply AM\geqslant GM relation

\begin{aligned} & \frac{a_{1}+a_{2}+a_{3}}{3} \geqslant\left(a_{1} \cdot a_{2} \cdot a_{3}\right)^{1 / 3} \\ \Rightarrow & \frac{a b}{c^{2}}+\frac{b c}{a^{2}}+\frac{c a}{b^{2}} \geqslant 3\left(\frac{a b}{c^{2}} \cdot \frac{b c}{a^{2}} \cdot \frac{c a}{b^{2}}\right)^{1 / 3} \\ \Rightarrow & \frac{a b}{c^{2}}+\frac{b c}{a^{2}}+\frac{c a}{b^{2}} \geqslant 3 \end{aligned}

So minium value is 3.

Posted by

vinayak

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