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216 7. Approximations and Inequalities

--y=at+b

3. Suppose that f(t) is a given function and that p(t) is a polynomial
   of degree not exceeding n. Let

K = max{ (f(t) - p(t)1 : a 5 t <_ b}.

Suppose that there are n + 2 distinct points tc,tr, t2,... , tn+l in the

interval such that

(i) If(ti) - p(ti)( = K (0 5 i 5 n + 1)

(ii) f(ti+l) - P(ti+l) = -[f(h) -P(h)] (0^5 i^5 n)

(i.e., the maximum distance between the graphs is achieved with al-

terna.te signs at least n + 2 times). The diagram illustrates a possible

situation when n = 3.

,‘--I

(a) Suppose that q(t) is a polynomial such that

max{lf(t) - q(t)] : a 5 t 5 b} < K.