Question Number 187651 by Mastermind last updated on 19/Feb/23
$$\mathrm{Find}\:\mathrm{the}\:\mathrm{range}\:\mathrm{of}\:\mathrm{this}\:\mathrm{function} \\ $$$$\mathrm{x}^{\mathrm{2}} \:−\mathrm{13x}\:+\:\mathrm{36}\:=\:\mathrm{0} \\ $$$$ \\ $$$$ \\ $$$$\mathrm{Help}! \\ $$
Answered by Ar Brandon last updated on 19/Feb/23
$$\mathrm{Let}\:\mathrm{y}={x}^{\mathrm{2}} −\mathrm{13}{x}+\mathrm{36} \\ $$$$\Rightarrow{x}^{\mathrm{2}} −\mathrm{13}{x}+\mathrm{36}−\mathrm{y}=\mathrm{0} \\ $$$$\Rightarrow{x}=\frac{\mathrm{13}\pm\sqrt{\mathrm{169}−\mathrm{4}\left(\mathrm{36}−\mathrm{y}\right)}}{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:\:\:=\frac{\mathrm{13}\pm\sqrt{\mathrm{25}+\mathrm{4y}}}{\mathrm{2}} \\ $$$$\mathrm{25}+\mathrm{4y}\:\geqslant\mathrm{0}\:\Rightarrow\mathrm{y}\geqslant−\frac{\mathrm{25}}{\mathrm{4}} \\ $$$$\mathrm{Hence}\:\mathrm{range}\:{f}\left({x}\right)\geqslant−\frac{\mathrm{25}}{\mathrm{4}} \\ $$
Commented by Mastermind last updated on 20/Feb/23
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{BOSS},\:\mathrm{that}'\mathrm{s}\:\mathrm{what}\:\mathrm{i}\:\mathrm{also} \\ $$$$\mathrm{got}\:\mathrm{but}\:\mathrm{with}\:\mathrm{diff}.\:\mathrm{method}. \\ $$
Answered by mr W last updated on 20/Feb/23
$${x}^{\mathrm{2}} −\mathrm{13}{x}+\mathrm{36} \\ $$$$=\left({x}−\frac{\mathrm{13}}{\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{36}−\left(\frac{\mathrm{13}}{\mathrm{2}}\right)^{\mathrm{2}} \\ $$$$\geqslant\mathrm{36}−\left(\frac{\mathrm{13}}{\mathrm{2}}\right)^{\mathrm{2}} =−\frac{\mathrm{25}}{\mathrm{4}} \\ $$$$\Rightarrow{range}\:{is}\:\left[−\frac{\mathrm{25}}{\mathrm{4}},+\infty\right) \\ $$