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Question Number 148676 by bemath last updated on 30/Jul/21

Find local minimum of function H(x)=((34)/(3+((2x)/(x^2 +3x+1)))) .

$$\mathrm{Find}\:\mathrm{local}\:\mathrm{minimum}\:\mathrm{of}\:\mathrm{function} \\ $$$$\:\mathrm{H}\left(\mathrm{x}\right)=\frac{\mathrm{34}}{\mathrm{3}+\frac{\mathrm{2x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{3x}+\mathrm{1}}}\:. \\ $$

Answered by EDWIN88 last updated on 30/Jul/21

H(x)=((34x^2 +102x+34)/(3x^2 +11x+3)) 3x^2 h+11hx+3h=34x^2 +102x+34 (34−3h)x^2 +(102−11h)x+34−3h=0 Δ≥0 (102−11h)^2 −4(34−3h)(34−3h)≥0 10404−2244h+121h^2 −4(1156−204h+9h^2 )≥0 121h^2 −2244h+10404−4624+816h−36h^2 ≥0 85h^2 −1428h+5780≥0 (h−10)(5h−34)≥0 h≤ ((34)/5)∪ h≥10 H(x)_(min) = 10 when x=1

$${H}\left({x}\right)=\frac{\mathrm{34}{x}^{\mathrm{2}} +\mathrm{102}{x}+\mathrm{34}}{\mathrm{3}{x}^{\mathrm{2}} +\mathrm{11}{x}+\mathrm{3}} \\ $$$$\mathrm{3}{x}^{\mathrm{2}} {h}+\mathrm{11}{hx}+\mathrm{3}{h}=\mathrm{34}{x}^{\mathrm{2}} +\mathrm{102}{x}+\mathrm{34} \\ $$$$\left(\mathrm{34}−\mathrm{3}{h}\right){x}^{\mathrm{2}} +\left(\mathrm{102}−\mathrm{11}{h}\right){x}+\mathrm{34}−\mathrm{3}{h}=\mathrm{0} \\ $$$$\Delta\geqslant\mathrm{0} \\ $$$$\left(\mathrm{102}−\mathrm{11}{h}\right)^{\mathrm{2}} −\mathrm{4}\left(\mathrm{34}−\mathrm{3}{h}\right)\left(\mathrm{34}−\mathrm{3}{h}\right)\geqslant\mathrm{0} \\ $$$$\mathrm{10404}−\mathrm{2244}{h}+\mathrm{121}{h}^{\mathrm{2}} −\mathrm{4}\left(\mathrm{1156}−\mathrm{204}{h}+\mathrm{9}{h}^{\mathrm{2}} \right)\geqslant\mathrm{0} \\ $$$$\mathrm{121}{h}^{\mathrm{2}} −\mathrm{2244}{h}+\mathrm{10404}−\mathrm{4624}+\mathrm{816}{h}−\mathrm{36}{h}^{\mathrm{2}} \geqslant\mathrm{0} \\ $$$$\mathrm{85}{h}^{\mathrm{2}} −\mathrm{1428}{h}+\mathrm{5780}\geqslant\mathrm{0} \\ $$$$\left({h}−\mathrm{10}\right)\left(\mathrm{5}{h}−\mathrm{34}\right)\geqslant\mathrm{0} \\ $$$${h}\leqslant\:\frac{\mathrm{34}}{\mathrm{5}}\cup\:{h}\geqslant\mathrm{10} \\ $$$${H}\left({x}\right)_{{min}} =\:\mathrm{10}\:{when}\:{x}=\mathrm{1}\: \\ $$

Commented by EDWIN88 last updated on 30/Jul/21

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