La solution d'exercice 1 - Exercices Corrigés d'Analyse II
La fonction
définie sur [0; 1] par
![](https://1.bp.blogspot.com/-0lIErS-Dkuo/Vx-qUOtzxNI/AAAAAAAAJ9c/0qkdM2Crj24gm1BOLTqoe1I9ItRFGiZ0ACLcB/s1600/image001.png)
![](https://4.bp.blogspot.com/-3GA-n9caOfo/Vx-qUOcsdBI/AAAAAAAAJ9g/_0pLbjymokMUXuwdPO_Mi4sy8Nj0qh0YQCLcB/s1600/image002.png)
est continue sur [0; 1[ et
![](https://2.bp.blogspot.com/-LuSta29NeXI/Vx-qUGh32kI/AAAAAAAAJ9k/MC0sxKCeofE0D1ahz-UcEHZV90jKPsBNACLcB/s1600/image003.png)
Elle set dérivable sur ]0; 1[ et
D'après le théorème de Rolle, il existe
tel que
. Or
![](https://3.bp.blogspot.com/-EsdRSCJyM_I/Vx-qUqMlFMI/AAAAAAAAJ9o/hvppgKqGTDI2B2qJ_OoT0gm_lPk5U58WQCLcB/s1600/image004.png)
![](https://1.bp.blogspot.com/-dgVp08wvP7U/Vx-qUpuvjEI/AAAAAAAAJ9s/yM3j2-JSb3cugpeWfQH6AoGx14P2bTF1ACLcB/s1600/image005.png)
![](https://3.bp.blogspot.com/-AF6ahjLQydM/Vx-qUu2ywUI/AAAAAAAAJ9w/XvzFf6jh5Q48wVvPOwD4DBNovgtLQkiyACLcB/s1600/image006.png)
![](https://2.bp.blogspot.com/-J-IiUfooCrA/Vx-qUzUT5TI/AAAAAAAAJ90/vext6WOV_jkBpPk9pt-eONbVSlWegB87ACLcB/s1600/image007.png)
Si
alors
.
![](https://1.bp.blogspot.com/-WY3hAXskvr4/Vx-qVINIHwI/AAAAAAAAJ94/lPcY-DGqLP0mhmcoSy7R3DakUDTswm5WQCLcB/s1600/image008.png)
![](https://3.bp.blogspot.com/-GELDDg9RkaI/Vx-qVI2dQgI/AAAAAAAAJ-A/aQI8YwK532Ul59a9CVDTrlwco7TFSLOXACLcB/s1600/image009.png)
1- Dessin : ………….
2- 2- On a
![](https://2.bp.blogspot.com/-s1-l8STmwnA/Vx-qVf1QXOI/AAAAAAAAJ98/ydM-BtnjbL07JKupYSE2QtGD9FPvKN4GQCLcB/s1600/image010.png)
Donc
![](https://3.bp.blogspot.com/-RQMSnYKQE34/Vx-qV6sf13I/AAAAAAAAJ-E/rHUowuk_PXY3BHv0vp1Eq6UEN5IP0xmvgCLcB/s1600/image011.png)
La fonction
est à valeurs dans R+ car arctan est croissante et
pour tout
. De plus elle est dérivable sur R* et
![](https://2.bp.blogspot.com/-5SOD4EWjRMU/Vx-qV9kooeI/AAAAAAAAJ-I/JhtZiEMrrYYV8F0I8chAzxO43siNEXCcgCLcB/s1600/image012.png)
![](https://2.bp.blogspot.com/-JqRJpA9p6N4/Vx-qV-8u3vI/AAAAAAAAJ-Q/dPtPqT0YNoUiFwlYfYTf-A8ZV2EPufGEQCLcB/s1600/image013.png)
![](https://2.bp.blogspot.com/-oBgiTYm1ybo/Vx-qWCz3dUI/AAAAAAAAJ-M/vYYtHsqbo9Q8wq2VTqK2hTpcD-Jq-fOQQCLcB/s1600/image014.png)
![](https://1.bp.blogspot.com/-XCSGjuwWgYQ/Vx-qWtvpL4I/AAAAAAAAJ-U/dMmFI9nprUE-kHJZFTHxy40JZC4_7m_EwCLcB/s1600/image015.png)
D'après ce qui précède , il existe un point
. Un calcul de la dérivée donne
![](https://3.bp.blogspot.com/-cc6_dVGPCrI/Vx-qW3BEGVI/AAAAAAAAJ-Y/1lPyIWP8uekkQSjvMW-WTuBLWOf5z1b3gCLcB/s1600/image016.png)
![](https://1.bp.blogspot.com/-wdo_JqJceZk/Vx-qWx1pXUI/AAAAAAAAJ-c/bC-WTXHgE3In5mp02w1pKjYoNmbpf8_gwCLcB/s1600/image017.png)
Donc
donne ![](https://4.bp.blogspot.com/-77DxFnkeJKk/Vx-qXEpAZZI/AAAAAAAAJ-o/nuhQKkcBVkoOCt-vgBRemBBbJd1FkYGygCLcB/s1600/image019.png)
![](https://4.bp.blogspot.com/-bAk_sx6K3z4/Vx-qXNyJZlI/AAAAAAAAJ-g/rw8vcz87kY07QPiJHjaqrI46ajVqIWY0gCLcB/s1600/image018.png)
![](https://4.bp.blogspot.com/-77DxFnkeJKk/Vx-qXEpAZZI/AAAAAAAAJ-o/nuhQKkcBVkoOCt-vgBRemBBbJd1FkYGygCLcB/s1600/image019.png)
Puisque c’est l'unique extremum de
sur ]0;+1[ et la fonction
est strictement positive sur ]0;
[, alors
est un maximum de
sur [0;
[ car
tend vers 0 quand d tend vers 0 et d tend vers
.
![](https://4.bp.blogspot.com/-Re9j_O9P-wU/Vx-qXcImELI/AAAAAAAAJ-k/oJ8yV1MN9loB0bdShYON_t-cqhQ6gsb6wCLcB/s1600/image020.png)
![](https://4.bp.blogspot.com/-Re9j_O9P-wU/Vx-qXcImELI/AAAAAAAAJ-k/oJ8yV1MN9loB0bdShYON_t-cqhQ6gsb6wCLcB/s1600/image020.png)
![](https://1.bp.blogspot.com/-mxJeSJkGlfU/Vx-qXk_QEeI/AAAAAAAAJ-s/CcVvv5ZcT5MYpsAEajS-t0eL2cKaf9qngCLcB/s1600/image021.png)
![](https://3.bp.blogspot.com/-fEblXcygf3Q/Vx-qXt5RHII/AAAAAAAAJ-w/C2JM9B-bEj8xLzX8G5Da_7OoKW-EWDZTgCLcB/s1600/image022.png)
![](https://4.bp.blogspot.com/-Re9j_O9P-wU/Vx-qXcImELI/AAAAAAAAJ-k/oJ8yV1MN9loB0bdShYON_t-cqhQ6gsb6wCLcB/s1600/image020.png)
![](https://1.bp.blogspot.com/-mxJeSJkGlfU/Vx-qXk_QEeI/AAAAAAAAJ-s/CcVvv5ZcT5MYpsAEajS-t0eL2cKaf9qngCLcB/s1600/image021.png)
![](https://2.bp.blogspot.com/-djCMTld18NQ/Vx-qXhxqilI/AAAAAAAAJ-4/RDB0DfxihEoBKuLjs2UYrSqQfgg3yjvEACLcB/s1600/image023.png)
![](https://1.bp.blogspot.com/-mxJeSJkGlfU/Vx-qXk_QEeI/AAAAAAAAJ-s/CcVvv5ZcT5MYpsAEajS-t0eL2cKaf9qngCLcB/s1600/image021.png)
Donc :
![](https://3.bp.blogspot.com/-H4CJgqTzH7o/Vx-qXy8KSjI/AAAAAAAAJ-0/UlkhGrKrvYQpEM2boV262mp4GoSS2Mf5QCLcB/s1600/image024.png)
Puisque
, alors l'angle maximum pour voir la statue d'une hauteur h est
![](https://3.bp.blogspot.com/-s0o22x9hdwk/Vx-qYdHQjEI/AAAAAAAAJ-8/9eKFgw31JNoJbymlpeNBiZzxnX9NFZ7AgCLcB/s1600/image025.png)
![](https://2.bp.blogspot.com/-D7PP7Lw_LJE/Vx-qYgbjBjI/AAAAAAAAJ_A/c4Ksi2T0tMQZVk7JwWJBr_YFgV3-PJ7MACLcB/s1600/image026.png)
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