Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0000 362488 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.0409 1211880 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.0409 1211880 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.1123 4049744 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.1137 4077440 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.1145 4095152 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.1147 4126184 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.1157 4148904 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.1157 4148904 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.1828 4779888 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.1857 4880872 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.1862 4901800 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.1963 4970592 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.1963 4970848 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

Lagrange Multipliers

Lagrange multipliers are used in optimising or maximising a functionsubject to one or more equality constraintsThe solution is found by minimizingwith respect to theandThe Lagrange multipliers are the

For example to maximisesubject toform the expression

Differentiating with respect togives (1)

Differentiating with respect togives (2)

Differentiating with respect togives (3)

From (1)

From (2)

From (3)

The maximum value ofis then

This is illustrated below. The maximum is where the curves touch.