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AM,GM,HM Inequality

For 0 < a <...

Question

For $0 < a < x$ , then the minimum value of the function ${log}_{a} x + {log}_{x} a$ is

1

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Solution

The correct option is B $2$
Given that:
${log}_{x} a + {log}_{a} x = \frac{log a}{log x} + \frac{log x}{log a} ∵ A . M \geq G . M ∴ \frac{1}{2} (\frac{log a}{log x} + \frac{log x}{log a}) \geq \sqrt{\frac{log a}{log x} \frac{log x}{log a}} \Rightarrow {log}_{x} a + {log}_{a} x \geq 2$

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\frac{d y}{d x}

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- 1 :

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a > 1

- 2 :

a > 1, {log}_{a} x

y + l o g_{a} x + l o g_{x} a + l o g_{a} a

\frac{d y}{d x} =

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