Please solve this questions below without ai and step by step.

$13\mathrm{.}2-2.$ Reconsider Prob. $13\mathrm{.}1-4.$ Show that the model formulated is a convex programming problem by using the test in Appendix $2$ to show that the objective function being minimized is convex.

$13\mathrm{.}2-3.$ Consider the variation of the Wyndor Glass Co$.$ example represented in Fig. $13\mathrm{.}5,$ where the second and third functional constraints of the original problem $($see Sec. $3\mathrm{.}1)$ have been replaced by $9x_1^2+5x_2^{2}216.$ Demonstrate that $(x_1,x_2)=(2,6)$ with $Z=36$ is indeed optimal by showing that the objective function line $36=3x_1+5x_2$ is tangent to this constraint boundary at $(2,6).($Hint: Express $x_2$ in terms of $x_1$ on this boundary, and then differentiate this expression with respect to $x_1$ to find the slope of the boundary.$)$

$13\mathrm{.}2-5.$ Consider the following function:

$f(x)=48x-60x^2+x^3$

$($a$)$ Use the first and second derivatives to find the local maxima and local minima of $f(x).$

$($b$)$ Use the first and second derivatives to show that $f(x)$ has neither a global maximum nor a global minimum because it is unbounded in both directions.

$13\mathrm{.}2-6.$ For each of the following functions, show whether it is convex, concave, or neither.

$($a$)f(x)=10x-x^2$

$($b$)f(x)=x^4+6x^2+12x$

$($c$)f(x)=2x^3-3x^2$

$($d$)f(x)=x^4+x^2$

$($e$)f(x)=x^3+x^4$

$13\mathrm{.}2-7.*$ For each of the following functions, use the test given in Appendix $2$ to determine whether it is convex, concave, or neither.

$($a$)f(x)=x_1{x}_2-x_1^2-x_2^2$

$($b$)f(x)=3x_1+2x_1^2+4x_2+x_2^2-2x_1{x}_2$

$($c$)f(x)=x_1^2+3x_1{x}_2+2x_2^2$

$($d$)f(x)=20x_1+10x_2$

$($e$)f(x)=x_1{x}_2$

$13\mathrm{.}2-8.$ Consider the following function:

$f(x)=5x_1+2x_2^2+x_3^2-3x_3{x}_4+4x_4^2+2x_5^4+x_5^2$

$+3x_5{x}_6+6x_6^2+3x_6{x}_7+x_7^2.$

Show that $f(x)$ is convex by expressing it as a sum of functions of one or two variables and then showing $($see Appendix $2)$ that all these functions are convex.