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andrei_g
Pe: 2013-05-13T18:53:23+03:00 In: In:

extreme

1. Gasiti valorile extreme ale expresiei E(x,y,z)=\[x+y+\frac{z}{3} pe multimea solutiilor sistemului

    \[\left\{ \begin{array}{l}2x + y + \cos ^3 z = 0 \\ x - y + \cos z = 0\quad ,x \in \left[ {0,\pi } \right] \\ \end{array} \right.\;\]

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2. Fie f:(-oo,0)->R, f(x)=x+b+c-3*rad de ord 3 din (xbc) ,unde b,c<0
a). aratati ca f'(x)<=0 , pt orice x <0
b).Deduceti inegalitatea (a+b+c)/3<=rad de ord 3 din abc ;oricare ar fi a,b,c <=0

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1 raspuns

  1. ghioknt
    profesor

    In primul rand imi permit sa scriu zE[0,pi], nu x.
    Rezolvand sitemul, aflam x si y in functie de cosz:

        \[ \begin{array}{l} x = \frac{{ - 1}}{3}\left( {\cos ^3 z + \cos z} \right),\,\,y = \frac{{ - 1}}{3}\left( {\cos ^3 z - 2\cos z} \right).\,\,obtinem\,\,E\left( {x,y,x} \right) = f\left( z \right) = \frac{1}{3}\left( {z - 2\cos ^3 z + \cos z} \right) \\ f'\left( z \right) = \frac{1}{3}\left( {1 + 6\cos ^2 z\sin z - \sin z} \right) = \frac{1}{3}\left( {1 + 5\sin z - 6\sin ^3 z} \right) = \frac{1}{3}\left( {1 - \sin z} \right)\left( {6\sin ^2 z + 6\sin z + 1} \right) \ge 0\,\,\forall z \in \left[ {0,\pi } \right] \\ pentru\,\,ca\,\,0 \le \sin z \le 1.\,\,Valoarea\,\,0\,\,este\,\,luata\,\,de\,\,f'\,\,{{\rm int}} r - un\,\,\sin gur\,\,punct,\,\,deci\,\,f\,\,este\,\,str.\,\,cresc. \\ \min \,E\left( {x,y,x} \right) = f\left( 0 \right) = \frac{{ - 1}}{3};\,\,\max \,E\left( {x,y,x} \right) = f\left( \pi \right) = \frac{{\pi + 1}}{3} \\ \end{array} \]

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    Cu bine, ghioknt.

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