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# Existence of constant (4)

Let be a continuous function such that

(1)

Prove that there exists a such that

Solution

For starters let us consider the function and . Trivially , it is and we note that:

Consider now the function

Clearly is continuous on and differentiable on with derivative

Furthermore , . Hence , by Rolle’s theorem there exists a such that ; that is . Finally, let us consider the function

which satisfies Rolle’s conditions on . Hence, there exists a such that . However,