Consider the cubic function g(x) = x3 − x − 1. Show that the three roots of g(x) = 0 are separated by ±1/ √ 3. Find the largest root. Take the derivative of f (x + tp) with respect to t to show that...

Consider the cubic function g(x) = x3 − x − 1. Show that the three roots of g(x) = 0 are separated by ±1/ √ 3. Find the largest root.

Take the derivative of f (x + tp) with respect to t to show that the directional derivative of a function f (x) in the direction p is equal to pT∇f (x).