$($a$)$ The minimum of a function may be bracketed by a triple $(a,b,c)$ where position $b$

lies between positions a and $c,$ and $b\frac{0\mathrm{.}61803}{0\mathrm{.}38197}f(b).$ The golden section

search $is$ one $of$ the more reliable methods employed $to$ locate a minimum $by$

progressively tightening the brackets bounding $it.$

Show, $by$ deriving a suitable equation, why the intermediate point $bis$ located $a$

fractional distance $of$ approximately $0\mathrm{.}38197$ from one end $(i.e.aorc),$ and

$0\mathrm{.}61803$ from the other; where the ratio $of$ the segments,$\frac{0\mathrm{.}61803}{0\mathrm{.}38197},is$

the golden ratio.