Science:Math Exam Resources/Courses/MATH100/December 2016/Question 07 (a)
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Question 07 (a) 

Let be a continuous function on the open interval . Which of the following four statements are always true?

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Hint 

Consider the intermediate value theorem. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. (A) True — by the intermediate value theorem. (B) False — the derivative of a function may not even exist at a local minimum! For example, consider on Although has its local minimum at does not exist. (C) False — just take the function on which has no minimum or maximum on (D) False — the derivative could be everywhere positive. For example, consider on While for any point in , it doesn't have local minimum on this interval. This is because is strictly increasing on the interval, i.e., The answer: 