Suppose that the continuous function...

Suppose that the continuous function f:[a,b]->R has the property that the integral form c to d is less than or equal to zero whenever a≤c
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Suppose not and let f(v)>0 where a v on which f is strictly positive, say on (x,y) where a But then integral of f from x to y is >0, a contradiction. It follows that
f(x)<=0 on [a,b].