By sketching a suitable pair of graphs on the same diagram, show that the equation
$$\mathrm { e } ^ { - \frac { 1 } { 2 } x } = x ^ { 5 }$$
has exactly one real root.
Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { \mathrm { e } ^ { - \frac { 1 } { 2 } x _ { n } } }\) to determine the root correct to 4 significant figures. Give the result of each iteration to 6 significant figures.