The Bernstein polynomial of degree n for f ∈ C[0, 1] is given by for each x...

The Bernstein polynomial of degree n for f ∈ C[0, 1] is given by

for each x ∈ [0, 1].

a. Find B₃(x) for the functions i. f (x) = x ii. f (x) = 1

b. Show that for each k ≤ n

c. Use part (b) and the fact, from (ii) in part (a), that

d. Use part (c) to estimate the value of n necessary for to hold for all x in [0, 1].