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# Integral function

Let be two continuous functions such that

If and for all , then prove that:

1. the equation has at least a root in if is differentiable.
2. the equation has at least a root in .
3. the area of bounded by the axis and the lines , is square meters.

Solution

1. Since for all it follows that the sign of is constant. For we get that hence for all . Therefore,

2. Taking limits we have