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Can anyone help me out. I reach to the conclusion that all three inequalities are wrong. But a) , b) are correct.(Answer is given in NBHM site)

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    Do you know about the Cauchy-Schwarz inequality? – 2017-01-19
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    yes.. But how it will be applied here????????? – 2017-01-19

1 Answers 1

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Hints:

For part (a), show that $\frac{1}{M}f(x)+m\frac{1}{f(x)}\ge 2\sqrt{\frac{m}{M}}$ for all $x$.

For part (b), notice that by Cauchy-Schwarz we have $$\int\limits_{a}^{b}{1\,dx} = \int\limits_{a}^{b}{\sqrt{f(x)}\frac{1}{\sqrt{f(x)}}\,dx}\le\left(\int\limits_{a}^{b}{f(x)\,dx}\right)^{1/2}\left(\int\limits_{a}^{b}{\frac{1}{f(x)}\,dx}\right)^{1/2}.$$

Furthermore, if you can find an example for which the inequality in part (b) is strict, then part (c) is false.

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    Thank You. It really helped me.. – 2017-01-19