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Question Number 74300 by ~blr237~ last updated on 21/Nov/19

Let consider  γ  :I→R^2   a parametric curve   1)Prove that if  a<b  and  γ(a)≠γ(b) then there exist  t_0 ∈]a,b[    such as  γ′(t_0 )  is colinear to γ(b)−γ(a)   2)Show that if  γ is regular and the  function f :I→R    t→f(t)=∣∣γ(t)−O(0,0) ∣∣  is maximal in t_0 ∈I  Then  ∣K_γ (t_0 )∣≥(1/(f(t_0 )))

$${Let}\:{consider}\:\:\gamma\:\::{I}\rightarrow\mathbb{R}^{\mathrm{2}} \:\:{a}\:{parametric}\:{curve}\: \\ $$ $$\left.\mathrm{1}\left.\right){Prove}\:{that}\:{if}\:\:{a}<{b}\:\:{and}\:\:\gamma\left({a}\right)\neq\gamma\left({b}\right)\:{then}\:{there}\:{exist}\:\:{t}_{\mathrm{0}} \in\right]{a},{b}\left[\:\:\right. \\ $$ $${such}\:{as}\:\:\gamma'\left({t}_{\mathrm{0}} \right)\:\:{is}\:{colinear}\:{to}\:\gamma\left({b}\right)−\gamma\left({a}\right)\: \\ $$ $$\left.\mathrm{2}\right){Show}\:{that}\:{if}\:\:\gamma\:{is}\:{regular}\:{and}\:{the}\:\:{function}\:{f}\::{I}\rightarrow\mathbb{R}\:\:\:\:{t}\rightarrow{f}\left({t}\right)=\mid\mid\gamma\left({t}\right)−{O}\left(\mathrm{0},\mathrm{0}\right)\:\mid\mid\:\:{is}\:{maximal}\:{in}\:{t}_{\mathrm{0}} \in{I} \\ $$ $${Then}\:\:\mid{K}_{\gamma} \left({t}_{\mathrm{0}} \right)\mid\geqslant\frac{\mathrm{1}}{{f}\left({t}_{\mathrm{0}} \right)} \\ $$

Answered by mind is power last updated on 21/Nov/19

γ(t)=(x(t),y(t))  γ(a)≠γ(b)  ⇒x(a)≠x(b)  or y(a)≠y(b)  x(a)≠x(b)  g(t)=y(t)(x(a)−x(b)−x(t)(y(a)−y(b))  g(a)=−y(a)x(b)+x(a)y(b)  g(b)=y(b)x(a)−x(b)y(a)=g(a)  g(a)=g(b) mean values ⇒∃t_0 ∈]a,b[∣g′(t_0 )=0  g′(t_0 )=y′(t_0 )(x(a)−x(b))−x′(t_0 ).(y(a)−y(b))=0  since γ(a)≠γ(b) we have 3 cases  if x(a)=x(b)⇒x′(t_0 )=0  x(a)≠x(b) & y(a)−y(b)=0⇒y′(t_0 )=0  x(a)#x(b)&y(a)≠y(b)⇒  ⇔((y′(t_0 ))/(x′(t_0 )))=((y(a)−y(b))/(x(a)−x(b)))  ((y′(t_0 ))/(x′(t_0 )))  is coeficuent of tangent in t_o   ⇒in t_0 ,γ′(t_0 )  is colinear to γ(b)−γ(a)  2) calcule  corbur

$$\gamma\left({t}\right)=\left({x}\left({t}\right),{y}\left({t}\right)\right) \\ $$ $$\gamma\left({a}\right)\neq\gamma\left({b}\right) \\ $$ $$\Rightarrow{x}\left({a}\right)\neq{x}\left({b}\right)\:\:{or}\:{y}\left({a}\right)\neq{y}\left({b}\right) \\ $$ $${x}\left({a}\right)\neq{x}\left({b}\right) \\ $$ $${g}\left({t}\right)={y}\left({t}\right)\left({x}\left({a}\right)−{x}\left({b}\right)−{x}\left({t}\right)\left({y}\left({a}\right)−{y}\left({b}\right)\right)\right. \\ $$ $${g}\left({a}\right)=−{y}\left({a}\right){x}\left({b}\right)+{x}\left({a}\right){y}\left({b}\right) \\ $$ $${g}\left({b}\right)={y}\left({b}\right){x}\left({a}\right)−{x}\left({b}\right){y}\left({a}\right)={g}\left({a}\right) \\ $$ $$\left.{g}\left({a}\right)={g}\left({b}\right)\:{mean}\:{values}\:\Rightarrow\exists{t}_{\mathrm{0}} \in\right]{a},{b}\left[\mid{g}'\left({t}_{\mathrm{0}} \right)=\mathrm{0}\right. \\ $$ $${g}'\left({t}_{\mathrm{0}} \right)={y}'\left({t}_{\mathrm{0}} \right)\left({x}\left({a}\right)−{x}\left({b}\right)\right)−{x}'\left({t}_{\mathrm{0}} \right).\left({y}\left({a}\right)−{y}\left({b}\right)\right)=\mathrm{0} \\ $$ $${since}\:\gamma\left({a}\right)\neq\gamma\left({b}\right)\:{we}\:{have}\:\mathrm{3}\:{cases} \\ $$ $${if}\:{x}\left({a}\right)={x}\left({b}\right)\Rightarrow{x}'\left({t}_{\mathrm{0}} \right)=\mathrm{0} \\ $$ $${x}\left({a}\right)\neq{x}\left({b}\right)\:\&\:{y}\left({a}\right)−{y}\left({b}\right)=\mathrm{0}\Rightarrow{y}'\left({t}_{\mathrm{0}} \right)=\mathrm{0} \\ $$ $${x}\left({a}\right)#{x}\left({b}\right)\&{y}\left({a}\right)\neq{y}\left({b}\right)\Rightarrow \\ $$ $$\Leftrightarrow\frac{{y}'\left({t}_{\mathrm{0}} \right)}{{x}'\left({t}_{\mathrm{0}} \right)}=\frac{{y}\left({a}\right)−{y}\left({b}\right)}{{x}\left({a}\right)−{x}\left({b}\right)} \\ $$ $$\frac{{y}'\left({t}_{\mathrm{0}} \right)}{{x}'\left({t}_{\mathrm{0}} \right)}\:\:{is}\:{coeficuent}\:{of}\:{tangent}\:{in}\:{t}_{{o}} \\ $$ $$\Rightarrow{in}\:{t}_{\mathrm{0}} ,\gamma'\left({t}_{\mathrm{0}} \right)\:\:{is}\:{colinear}\:{to}\:\gamma\left({b}\right)−\gamma\left({a}\right) \\ $$ $$\left.\mathrm{2}\right)\:{calcule}\:\:{corbur} \\ $$ $$ \\ $$ $$ \\ $$

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