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4-x-2-2-9-x-2-2-min-




Question Number 176893 by infinityaction last updated on 27/Sep/22
    [4(x^2 +2) +(9/((x^2 +2)))]_(min)
$$\:\:\:\:\left[\mathrm{4}\left({x}^{\mathrm{2}} +\mathrm{2}\right)\:+\frac{\mathrm{9}}{\left({x}^{\mathrm{2}} +\mathrm{2}\right)}\right]_{\mathrm{min}} \\ $$
Answered by mr W last updated on 27/Sep/22
t=x^2 +2≥2  4t+(9/t) is increasing for t≥(3/2).  (4t+(9/t))_(min) =4×2+(9/2)=((25)/2)=12.5
$${t}={x}^{\mathrm{2}} +\mathrm{2}\geqslant\mathrm{2} \\ $$$$\mathrm{4}{t}+\frac{\mathrm{9}}{{t}}\:{is}\:{increasing}\:{for}\:{t}\geqslant\frac{\mathrm{3}}{\mathrm{2}}. \\ $$$$\left(\mathrm{4}{t}+\frac{\mathrm{9}}{{t}}\right)_{{min}} =\mathrm{4}×\mathrm{2}+\frac{\mathrm{9}}{\mathrm{2}}=\frac{\mathrm{25}}{\mathrm{2}}=\mathrm{12}.\mathrm{5} \\ $$

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