$\mathrm{restart}$

$\mathrm{with}\left(\mathrm{Optimization}\right)\:$$\mathrm{with}\left(\mathrm{plots}\right)\:$

$\mathrm{obj}:=-2x-y$

$\mathrm{obj}≔-2x-y$

$\mathrm{cnsts}:=\left[y\le 4x\+\frac{1}{2}\,y\le -5x\+2\,0\le x\,0\le y\right]$

$\mathrm{cnsts}≔\left[y\le 4x+\frac{1}{2}\,y\le -5x+2\,0\le x\,0\le y\right]$

$\mathrm{p1}:=\mathrm{inequal}\left(\mathrm{cnsts}\,x\=-0.5..2\,y\=-0.5..2\,\mathrm{optionsexcluded}\=\left(\mathrm{color}\=\mathrm{white}\right)\,\mathrm{optionsfeasible}\=\left(\mathrm{color}\=\mathrm{yellow}\right)\right)\:$$\mathrm{p2}:=\mathrm{contourplot}\left(\mathrm{obj}\,x\=-0.5..2\,y\=-0.5..2\right)\:$$\mathrm{p3}≔\mathrm{pointplot}\left(\left\{\left[0.1666\,1.166\right]\right\}\,\mathrm{symbolsize}\=13\,\mathrm{color}\=\mathrm{green}\right)\:$$\mathrm{display}\left(\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right)$

$\mathrm{LPSolve}\left(\mathrm{obj}\,\mathrm{cnsts}\right)$

$\left[\mathrm{-1.50000000000000}\,\left[x=0.166666666666667\,y=1.16666666666667\right]\right]$

$\mathrm{LPSolve}\left(\mathrm{obj}\,\mathrm{cnsts}\left[1..2\right]\,\mathrm{assume}\=\mathrm{nonnegative}\right)$

$\left[\mathrm{-1.50000000000000}\,\left[x=0.166666666666667\,y=1.16666666666667\right]\right]$

$\mathrm{obj}:=2x\+y\+z$

$\mathrm{obj}≔2x+y+z$

$\mathrm{cnsts}:=\left[y\le 4x\+\frac{1}{2}\,y\le -5x\+2\,z\=-\frac{2x}{3}-\frac{2y}{3}\+\frac{20}{3}\right]$

$\mathrm{cnsts}≔\left[y\le 4x+\frac{1}{2}\,y\le -5x+2\,z=-\frac{2x}{3}-\frac{2y}{3}+\frac{20}{3}\right]$

$\mathrm{LPSolve}\left(\mathrm{obj}\,\mathrm{cnsts}\,\mathrm{assume}\=\mathrm{nonnegative}\,\mathrm{maximize}\right)$

$\left[7.27777777777778\,\left[x=0.166666666666667\,y=1.16666666666667\,z=5.77777777777778\right]\right]$

$\mathrm{obj}:=4x^2-y\+x\+5$

$\mathrm{obj}≔4x^2+x-y+5$

$\mathrm{cnsts}:=\left[-11\le x\+y-7x\,\frac{11x}{2}\+y\le 0\,-4\le x\right]$

$\mathrm{cnsts}≔\left[\mathrm{-11}\le -6x+y\,\frac{11x}{2}+y\le 0\,\mathrm{-4}\le x\right]$

$\mathrm{p1}:=\mathrm{contourplot}\left(\mathrm{obj}\,x\=-4..4\,y\=-10..10\,\mathrm{contours}\=30\right)\:$$\mathrm{p2}:=\mathrm{inequal}\left(\mathrm{cnsts}\,x\=-4..4\,y\=-10..10\,\mathrm{optionsexcluded}\=\left(\mathrm{color}\=\mathrm{white}\right)\,\mathrm{optionsfeasible}\=\left(\mathrm{color}\=\mathrm{yellow}\right)\right)\:$$\mathrm{p3}≔\mathrm{pointplot}\left(\left\{\left[-0.8125\,4.486\right]\right\}\,\mathrm{symbolsize}\=13\,\mathrm{color}\=\mathrm{green}\right)\:$$\mathrm{display}\left(\mathrm{p1}\,\mathrm{p2}\,\mathrm{p3}\right)$

$\mathrm{QPSolve}\left(4x^2-y\+x\+5\,\left\{-11\le x\+y-7x\,\frac{11x}{2}\+y\le 0\,-10\le x\right\}\right)$

$\left[2.35937499999999\,\left[x=\mathrm{-0.812500000000001}\,y=4.46875000000001\right]\right]$

$p:=0.004126984123x^7-0.1091666666x^6\+7.078095236x\+1.136388888x^5-18.19500000x^2-5.845833331x^4\+15.23138888x^3\+2$

$p≔0.004126984123x^7-0.1091666666x^6+7.078095236x+1.136388888x^5-18.19500000x^2-5.845833331x^4+15.23138888x^3+2$

$\mathrm{p1}:=\mathrm{plot}\left(p\,x\=0..6\right)\:$$\mathrm{p2}:=\mathrm{pointplot}\left(\left\{\left[4.8877\,0.6883\right]\right\}\,\mathrm{symbolsize}\=13\,\mathrm{color}\=\mathrm{green}\right)\:$$\mathrm{display}\left(\mathrm{p1}\,\mathrm{p2}\right)$

$\mathrm{NLPSolve}\left(p\,x\=0..6\right)$

$\left[0.683344558794943\,\left[x=4.88774385576649\right]\right]$

$\mathrm{NLPSolve}\left(p\,x\=0..6\,\mathrm{maximize}\right)$

$\left[2.80062108007189\,\left[x=2.97993884682345\right]\right]$

$f:=x\mathrm{erf}\left(x\right)\cos \left(x\right)$

$f≔x\mathrm{erf}\left(x\right)\cos \left(x\right)$

$s:=\mathrm{NLPSolve}\left(f\,x\=1..20\right)$

$s≔\left[\mathrm{-9.47729425947979}\,\left[x=9.52933438691188\right]\right]$

$\mathrm{p1}:=\mathrm{plot}\left(f\,x\=1..20\right)\:$$\mathrm{p2}:=\mathrm{pointplot}\left(\left[\left[\mathrm{rhs}\left(s\left[2\right]\left[1\right]\right)\,s\left[1\right]\right]\right]\,\mathrm{color}\=\mathrm{green}\,\mathrm{symbolsize}\=14\right)\:$$\mathrm{display}\left(\mathrm{p1}\,\mathrm{p2}\right)$

$f:=100{\left(y-x^2\right)}^2\+{\left(1-x\right)}^2$

$f≔100{\left(-x^2+y\right)}^2+{\left(1-x\right)}^2$

$\mathrm{NLPSolve}\left(f\,x\=-10..10\,y\=-10..10\,\mathrm{initialpoint}\=\left[x\=-1.2\,y\=1\right]\right)$

$\left[1.79319068453110564\times {10}^{\mathrm{-16}}\,\left[x=0.999999988744863\,y=0.999999976764186\right]\right]$

$\mathrm{restart}$$\mathrm{with}\left(\mathrm{Optimization}\right)\:$

$\mathrm{data}:=\left[\left[1\,1.5\right]\,\left[2\,3.5\right]\,\left[2.5\,1.9\right]\,\left[3.1\,4.5\right]\,\left[4.3\,1.9\right]\,\left[4.7\,2.4\right]\,\left[5.8\,3\right]\,\left[6\,5.7\right]\,\left[6.6\,3.4\right]\,\left[6.7\,0.5\right]\,\left[7.1\,4\right]\,\left[7.4\,1.7\right]\right]$

$\mathrm{data}≔\left[\left[1\,1.5\right]\,\left[2\,3.5\right]\,\left[2.5\,1.9\right]\,\left[3.1\,4.5\right]\,\left[4.3\,1.9\right]\,\left[4.7\,2.4\right]\,\left[5.8\,3\right]\,\left[6\,5.7\right]\,\left[6.6\,3.4\right]\,\left[6.7\,0.5\right]\,\left[7.1\,4\right]\,\left[7.4\,1.7\right]\right]$

$p:=a{x}^5\+b{x}^4\+c{x}^3\+d{x}^2\+ex\+f$

$p≔a{x}^5+b{x}^4+c{x}^3+d{x}^2+ex+f$

$\mathrm{residues}:=\mathrm{map}\left(d\→\genfrac{}{}{0ex}{}{p}{\phantom{x\=d_1}}|\genfrac{}{}{0ex}{}{\phantom{p}}{x\=d_1}-d_2\,\mathrm{data}\right)$

$\mathrm{residues}≔\left[a+b+c+d+e+f-1.5\,32a+16b+8c+4d+2e+f-3.5\,97.65625a+39.0625b+15.625c+6.25d+2.5e+f-1.9\,286.29151a+92.3521b+29.791c+9.61d+3.1e+f-4.5\,1470.08443a+341.8801b+79.507c+18.49d+4.3e+f-1.9\,2293.45007a+487.9681b+103.823c+22.09d+4.7e+f-2.4\,6563.56768a+1131.6496b+195.112c+33.64d+5.8e+f-3\,7776a+1296b+216c+36d+6e+f-5.7\,12523.32576a+1897.4736b+287.496c+43.56d+6.6e+f-3.4\,13501.25107a+2015.1121b+300.763c+44.89d+6.7e+f-0.5\,18042.29351a+2541.1681b+357.911c+50.41d+7.1e+f-4\,22190.06624a+2998.6576b+405.224c+54.76d+7.4e+f-1.7\right]$

$\mathrm{sol}:=\mathrm{LSSolve}\left(\mathrm{residues}\right)$

$\mathrm{sol}≔\left[9.79941650829796096\,\left[a=0.00506817808955152\,b=\mathrm{-0.148717633940652}\,c=1.53722867786023\,d=\mathrm{-7.04487080570615}\,e=14.2425902266783\,f=\mathrm{-7.12139344171469}\right]\right]$

$\mathrm{poly}:=\mathrm{eval}\left(p\,\mathrm{sol}\left[2\right]\right)$

$\mathrm{poly}≔0.00506817808955152x^5-0.148717633940652x^4+1.53722867786023x^3-7.04487080570615x^2+14.2425902266783x-7.12139344171469$

$\mathrm{with}\left(\mathrm{plots}\right)\:$$\mathrm{p1}≔\mathrm{pointplot}\left(\mathrm{data}\right)\:$$\mathrm{p2}≔\mathrm{plot}\left(\mathrm{poly}\,x\=0.9..8\right)\:$$\mathrm{display}\left(\mathrm{p1}\,\mathrm{p2}\right)$