Answer
(b) $\max x_{k}=\{2,1,3,1\}=3$
$(\mathrm{a}) \sum_{k=1}^{4} \Delta x_{k}^{*} f\left(x_{k}^{*}\right) =-\frac{117}{16}$
Work Step by Step
We find:
$\sum_{k=1}^{4} \Delta x_{k}^{*} f\left(x_{k}^{*}\right) =\Delta x_{1}^{*} f\left(x_{1}^{*}\right) +\Delta x_{2}^{*} f\left(x_{2}^{*}\right) +\Delta x_{3}^{*} f\left(x_{3}^{*}\right) +\Delta x_{4}^{*} f\left(x_{4}^{*}\right) $
$=1\cdot f\left(-\frac{5}{2}\right) +2\cdot f(-1) +2\cdot f\left(\frac{1}{4}\right) +3 \cdot f(3) $
$=1\cdot-\frac{9}{4} +2 \cdot 3+1\cdot \frac{63}{16} -3 \cdot 5$
$=-\frac{117}{16}$
(b) $\max x_{k}=\{2,1,3,1\}=3$
