Take the equation f (x) x 2x2 1 4 x4. Consider the problem of finding

Take the equation f (x) Æ x ¡2x2 Å 1 4 x4. Consider the problem of finding the minimum over x ¸ 1.

(a) For what values of x is f (x) convex?

(b) Find the Newton iteration rule xi !xiÅ1.

(c) Using the starting value x1 Æ 1 calculate the Newton iteration x2.

(d) Consider Golden Section search. Start with the bracket [a,b] Æ [1,5]. Calculate the intermediate points c and d.

(e) Calculate f (x) at a,b,c,d. Does the function satisfy f

(a) È f

(c) and f

(d) Ç f (b)?

(f ) Given these calculations, find the updated bracket.