Question Number 221036 by SdC355 last updated on 23/May/25
![prove ∫_(P∈[ε,ε+δ]) (√((((d )/dt)f(t))^2 +(((d )/dt)g(t))^2 )) dt≤∫_(P∈[ε,ε+δ]) ((C_1 f^((1)) (t)+C_2 g^((1)) (t))/( (√(C_1 ^2 +C_2 ^2 )))) dt](https://www.tinkutara.com/question/Q221036.png)
$$\mathrm{prove} \\ $$$$\int_{{P}\in\left[\epsilon,\epsilon+\delta\right]} \sqrt{\left(\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}{f}\left({t}\right)\right)^{\mathrm{2}} +\left(\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\mathrm{g}\left({t}\right)\right)^{\mathrm{2}} }\:\mathrm{d}{t}\leq\int_{{P}\in\left[\epsilon,\epsilon+\delta\right]} \:\frac{{C}_{\mathrm{1}} {f}^{\left(\mathrm{1}\right)} \left({t}\right)+{C}_{\mathrm{2}} \mathrm{g}^{\left(\mathrm{1}\right)} \left({t}\right)}{\:\sqrt{{C}_{\mathrm{1}} ^{\mathrm{2}} +{C}_{\mathrm{2}} ^{\mathrm{2}} }}\:\mathrm{d}{t} \\ $$