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Question Number 175871 by infinityaction last updated on 08/Sep/22

f(x) = (√(x^2 −4x+13)) + (√(x^2 −14x+130)) minimum value of f(x) x ∈ R

$$\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{4x}+\mathrm{13}}\:+\:\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{14x}+\mathrm{130}} \\ $$$$\:\mathrm{minimum}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{f}\left(\mathrm{x}\right)\:\:\mathrm{x}\:\in\:\mathbb{R}\: \\ $$

Answered by mr W last updated on 08/Sep/22

f(x)=(√((x−2)^2 +3^2 ))+(√((7−x)^2 +9^2 )) ≥(√((x−2+7−x)^2 +(3+9)^2 ))=(√(5^2 +12^2 ))=13 ⇒f(x)_(min) =13

$${f}\left({x}\right)=\sqrt{\left({x}−\mathrm{2}\right)^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} }+\sqrt{\left(\mathrm{7}−{x}\right)^{\mathrm{2}} +\mathrm{9}^{\mathrm{2}} } \\ $$$$\geqslant\sqrt{\left({x}−\mathrm{2}+\mathrm{7}−{x}\right)^{\mathrm{2}} +\left(\mathrm{3}+\mathrm{9}\right)^{\mathrm{2}} }=\sqrt{\mathrm{5}^{\mathrm{2}} +\mathrm{12}^{\mathrm{2}} }=\mathrm{13} \\ $$$$\Rightarrow{f}\left({x}\right)_{{min}} =\mathrm{13} \\ $$

Answered by cortano1 last updated on 08/Sep/22

f(x)=(√((2−x)^2 +9)) +(√((x−7)^2 +81)) f(x) min when ((2−x)/(x−7)) = (3/9) ((2−x)/(x−7)) =(1/3) ⇒x−7=6−3x x=((13)/4) ⇒min f(x)=(√((2−((13)/4))^2 +9)) +(√((((13)/4)−7)^2 +81)) = (√(((25)/(16))+9)) +(√(((225)/(16))+81)) =(√((169)/(16))) +(√((1521)/(16))) =13

$$\:\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\left(\mathrm{2}−\mathrm{x}\right)^{\mathrm{2}} +\mathrm{9}}\:+\sqrt{\left(\mathrm{x}−\mathrm{7}\right)^{\mathrm{2}} +\mathrm{81}} \\ $$$$\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{min}\:\mathrm{when}\:\frac{\mathrm{2}−\mathrm{x}}{\mathrm{x}−\mathrm{7}}\:=\:\frac{\mathrm{3}}{\mathrm{9}} \\ $$$$\:\:\:\frac{\mathrm{2}−\mathrm{x}}{\mathrm{x}−\mathrm{7}}\:=\frac{\mathrm{1}}{\mathrm{3}}\:\Rightarrow\mathrm{x}−\mathrm{7}=\mathrm{6}−\mathrm{3x} \\ $$$$\:\:\mathrm{x}=\frac{\mathrm{13}}{\mathrm{4}}\:\Rightarrow\mathrm{min}\:\mathrm{f}\left(\mathrm{x}\right)=\sqrt{\left(\mathrm{2}−\frac{\mathrm{13}}{\mathrm{4}}\right)^{\mathrm{2}} +\mathrm{9}}\:+\sqrt{\left(\frac{\mathrm{13}}{\mathrm{4}}−\mathrm{7}\right)^{\mathrm{2}} +\mathrm{81}} \\ $$$$\:\:=\:\sqrt{\frac{\mathrm{25}}{\mathrm{16}}+\mathrm{9}}\:+\sqrt{\frac{\mathrm{225}}{\mathrm{16}}+\mathrm{81}} \\ $$$$\:\:\:=\sqrt{\frac{\mathrm{169}}{\mathrm{16}}}\:+\sqrt{\frac{\mathrm{1521}}{\mathrm{16}}}\:=\mathrm{13} \\ $$

Commented by infinityaction last updated on 08/Sep/22

explain your 2^(nd) line this ((2−x)/(x−7)) = (3/9)

$$\mathrm{explain}\:\mathrm{your}\:\mathrm{2}^{\mathrm{nd}} \:\mathrm{line} \\ $$$$\mathrm{this}\:\:\frac{\mathrm{2}−{x}}{{x}−\mathrm{7}}\:\:=\:\frac{\mathrm{3}}{\mathrm{9}} \\ $$

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