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Question Number 186962 by mnjuly1970 last updated on 12/Feb/23

Answered by MJS_new last updated on 12/Feb/23

$$\sqrt{a}=p\wedge \sqrt{b}=q\Rightarrow p,q>0$$$$(\frac{1}{p^2}-q-1)(\frac{1}{q^2}-p-1)⩾\frac{25}{4}$$$$q=1-p\Rightarrow 0<p<1$$$$(\frac{1}{p^2}-(1-p)-1)(\frac{1}{{(1-p)}^2}-p-1)⩾\frac{25}{4}$$$$\frac{{(p^2-p-1)}^2}{p(1-p)}⩾\frac{25}{4}$$$$\min \left(\mathrm{lhs}\right)=\frac{25}{4}\mathrm{at}p=\frac{1}{2}$$

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