Question Number 210587 by mnjuly1970 last updated on 13/Aug/24
$$ \\ $$$$\:\:\:\:\:{f}\left({x}\right)=\:\sqrt{\:\mathrm{13}\:−\mathrm{12}\sqrt{{x}}\:\:}\:+\:\sqrt{\mathrm{25}\:−\mathrm{24}\sqrt{\mathrm{1}−{x}}\:} \\ $$$$ \\ $$$$\:\:\:\:\:\:{find}\::\:\:\:\:\mathrm{M}{in}\:\left(\:{f}\:\right)=? \\ $$$$ \\ $$$$ \\ $$
Commented by Frix last updated on 13/Aug/24
$$\mathrm{I}\:\mathrm{get}\:\mathrm{2}\sqrt{\mathrm{5}} \\ $$
Commented by mnjuly1970 last updated on 13/Aug/24
$${yes}\:{sir}….\mathrm{2}\sqrt{\mathrm{5}}\:{is}\:{correct}. \\ $$
Commented by Ghisom last updated on 14/Aug/24
$${f}\left({x}\right)=\sqrt{\mathrm{2}{p}^{\mathrm{2}} +\mathrm{2}{p}+\mathrm{1}−\mathrm{2}{p}\left({p}+\mathrm{1}\right)\sqrt{{x}}}+\sqrt{\mathrm{2}{p}^{\mathrm{2}} +\mathrm{6}{p}+\mathrm{5}−\mathrm{2}\left({p}+\mathrm{1}\right)\left({p}+\mathrm{2}\right)\sqrt{\mathrm{1}−{x}}} \\ $$$${p}\geqslant\mathrm{0} \\ $$$$\Rightarrow \\ $$$$\mathrm{min}\:\left({f}\left({x}\right)\right)\:=\sqrt{\mathrm{2}\left({p}^{\mathrm{2}} +\mathrm{2}{p}+\mathrm{2}\right)} \\ $$$$\mathrm{at}\:{x}=\frac{{p}^{\mathrm{2}} \left({p}^{\mathrm{3}} +\mathrm{5}{p}^{\mathrm{2}} +\mathrm{12}{p}+\mathrm{10}−\left({p}+\mathrm{2}\right)^{\mathrm{2}} \sqrt{{p}^{\mathrm{4}} +\mathrm{4}{p}^{\mathrm{3}} +\mathrm{10}{p}^{\mathrm{2}} +\mathrm{12}{p}+\mathrm{4}}\right)}{\mathrm{2}\left({p}+\mathrm{1}\right)^{\mathrm{2}} \left({p}^{\mathrm{2}} +\mathrm{2}{p}+\mathrm{2}\right)^{\mathrm{2}} } \\ $$