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Numerade Educator

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Problem 91 Hard Difficulty

Suppose $ f $ is differentiable on an interval $ l $ and $ f'(x) > 0 $ for all numbers $ x $ in $ l $ except for a single number $ c $. Prove that $ f $ is increasing on the entire interval $ l $.

Answer

decreasing behavior

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Video Transcript

okay. As we can see, the function is continuous. This is important because this means it doesn't change and the derivative is always equivalent to zero. We also know that the function grows because the first derivative is greater than zero. And we know that when the derivative is equivalent to zero, then approaches zero from the right hand side from a positive value. And then we know there's a point of infection where the graft changes con cavity. So the solution to this would be decreasing behavior and the given interval.