The function is given as,

Differentiating the given function w.r.t. x,

Calculating the critical points,

Calculating the roots of the given quadratic equation,
![\begin{gathered} x\text{ = }\frac{-(-63)\pm\sqrt[]{(-63)^2-4*3*312}}{2*3} \\ x\text{ = }13\text{ and x = 8} \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/w6oya0gdws6q2ofjpsqgaf6mvd6s1pvtir.png)
The interval is given as,

Minimum value at x = 3 is calculated as,

Minimum value at x = 8 is calculated as,

Maximum value at x = 15,

Maximum value at x = 13 is calculated as,

Thus the required answer is,
