The function is given as,
[tex]E(x)\text{ = }2x^3-63x^2+624x\text{ + 816}[/tex]Differentiating the given function w.r.t. x,
[tex]E^{\prime}(x)=6x^2-126x+\text{ 624}[/tex]Calculating the critical points,
[tex]\begin{gathered} E^{\prime}(x)\text{ = 0} \\ 6x^2-126x+\text{ 624 = 0} \\ 3x^2-63x+312\text{ = 0} \end{gathered}[/tex]Calculating the roots of the given quadratic equation,
[tex]\begin{gathered} x\text{ = }\frac{-(-63)\pm\sqrt[]{(-63)^2-4\times3\times312}}{2\times3} \\ x\text{ = }13\text{ and x = 8} \end{gathered}[/tex]The interval is given as,
[tex]3\leq x\leq15[/tex]Minimum value at x = 3 is calculated as,
[tex]\begin{gathered} E(3)\text{ = }2(3)^3-63(3)^2+624(3)\text{ + 816} \\ E(3)\text{ = }2\times\text{ 8 - 63 }\times\text{ 9 + 624}\times3+816 \\ E(3)\text{ = }16\text{ - 567+1872 + 816} \\ E(x)\text{ = 2137} \end{gathered}[/tex]Minimum value at x = 8 is calculated as,
[tex]\begin{gathered} E(8)\text{ = }2(8)^3-63(8)^2+624(8)\text{ + 816} \\ E(8)\text{ =}1024\text{ - 4032 + 4992 + 816} \\ E(8)\text{ = 2800} \end{gathered}[/tex]Maximum value at x = 15,
[tex]\begin{gathered} E(15)\text{ = }2(15)^3-63(15)^2+624(15)\text{ + 816} \\ E(15)\text{ = }6750\text{ - 14175 + 9360 + 816} \\ E(15)\text{ =}2751 \end{gathered}[/tex]Maximum value at x = 13 is calculated as,
[tex]\begin{gathered} E(13)\text{ = }2(13)^3-63(13)^2+624(13)\text{ + 816} \\ E(13)\text{ = }4394\text{ - 10647 + 8112 + 816 } \\ E(13)\text{ = }2675 \end{gathered}[/tex]Thus the required answer is,
[tex]\begin{gathered} Absolute\text{ minimum : E(3) = 2137} \\ \text{Absolute max imum : E}(8)\text{ = 2800} \end{gathered}[/tex]