The function f ( x ) = cos( πx ) for x ∈ [0 , 1] is approximated by the following piecewise polynomial p of degree ≤ 3. The interval [0 , 1] is divided into k subintervals [ a j − 1 ,a j ] with a j = j/k for j = 0 , 1 ,...,k , and in each subinterval p is the polynomial interpolating f at 4 equally spaced points: a j − 1 , a j − 1 + 1 / (3 k ), a j − 1 + 2 / (3 k ), and a j . How should k be chosen so that the error max 0 ≤ x ≤ 1 | f ( x ) − p ( x ) |≤ 10 − 8 ? That is, derive an error upper bound, find the magnitude of | f (4) ( x ) | , and next get the desired value of k .