Question: Consider a continuous, positive function f: . The graph of f(x) is the set of points in the (x , y) Cartesian plane such that y= f(x) (See Figure 1).

Figure 1 : The graph of f(x)

Part a

Consider the area A under the graph of f(x), for a x b (See Figure 2).

Figure 2: Area under the graph of f(x)

Define

m = min{ f(x) : a x b }

and

M = max{ f(x) : a x b}

Note that m is a Minimum and M is a Maximum. Argue in simple geometric terms that

m(b-a) A M (b-a). (12)

In fact, it is enough for you to draw a picture that makes this fact evident.

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