Ex$.1\mathrm{.}6.$ Consider the scalar bounded nonlinear function

Maximise $f(x)=\{\begin{array}{cc}{x}^2,& 0x2\\ \frac{18-x}{4},& 2x3.\end{array}$

Give the MATLAB commands for solving the above problem using fminbnd.

Perform two iterations of the dichotomous search to determine the interval of uncertainty. Assume that $=0\mathrm{.}1.$

Ex$.1\mathrm{.}7.$ Consider the following scalar nonlinear function, which is being solved by the dichotomous search with $=0\mathrm{.}1,$

minf$(x)=\{\begin{array}{cc}4(x-1{)}^2,& 0x2,\\ 2x,& 2x3.\end{array}$

Perform two iterations to determine the interval of uncertainty.

Write a MATLAB code for solving the problem such that the absolute error is less than ${10}^{-6}.$ The code should output the approximated optimal point.

Give the MATLAB commands for solving the problem using fminbnd.

Ex$.1\mathrm{.}8.$ Consider the scalar nonlinear function

maxf$(x)=\{\begin{array}{ccc}x+1,& \text{x}\text{i}\text{n}[-2,& -1],\\ 1-x^2,& \text{x}\text{i}\text{n}[-1,& 1].\end{array}$

Perform two iterations of the golden section method to determine the interval of uncertainty.

Give the MATLAB commands for solving the above problem using fminbnd.