$1.$ Design a circular tank closed at both ends to have a volume of $\backslash (250\backslash$mathrm$\{$~m$\}^\{3\}\backslash ).$ The fabrication cost is proportional to the surface area of the sheet metal and is $\backslash (\backslash$$ $400/\backslash$mathrm$\{$m$\}^\{2\}\backslash ).$ The tank is to be housed in a shed with a sloping roof. Therefore, height $\backslash ($ H $\backslash )$ of the tank is limited by the relation $\backslash ($ H $\backslash$leq$(10-$D $/2)\backslash ),$ where $\backslash ($ D $\backslash )$ is the tank$\text{'}$s diameter. Formulate the minimum$-$cost design problem in standard form. $(30$ pts $.)$

$2.$ Check the convexity of the problem formulated in Question $1.$ Explain your result. $(15$ pts$.)$

$3.$ Find points satisfying KKT necessary conditions for the problem formulated in Question $1.$ If there are global and local extremums, indicate them. Explain your result. $(35$ pts$.)$

$4.$ In the Matlab environment, create a graphical representation of the problem formulated in Question $1$ to verify your result obtained in Question $3.$ Indicate and mark the active and inactive constraints and local and global extremums if they exist on the graph. Investigate and explain the graph. $(20$ pts$.)$