Suppose P = {x0,x1,x2, ··· ,xn} is a partition of f (x). Consider the partition Q = {x0,x1, ··· ,xk−1, t,xk, ··· ,xn}.
Notice that Q is simply the partition P but with just one more point added. It should make sense that Lf (P) <= Lf (Q). Let’s prove this formally.

(a) Let m' = min{ f (x) : xk−1 <= x <= t} and m'' = min{ f (x) : t <= x <= xk}. Prove that Lf (P) <= Lf (Q) if and
only if mk(xk −xk−1) <= m' (t −xk−1)+m'' (xk −t).
(b) Now prove that mk(xk −xk−1) <= m'(t −xk−1)+m''(xk −t), thereby completing the proof that Lf (P) <= Lf (Q). (What can you say about mk compared to m', and mk compared to m''?)
(c) Let P and Q be the partitions as given above. What can you say about the upper sums? Deduce a similar
statement and give a proof.

Notes: m' = first derivative of m. m'' = second derivative of m. xk = x sub k.

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