Science:Math Exam Resources/Courses/MATH102/December 2016/Question A 08
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Question A 08 

Which of the following statements is correct? (i) Newton’s method finds the zero closest to the initial guess . (ii) The function has a tangent line at some that goes through the point . (iii) A tangent line to the graph of at an inflection point must cross from one side to the other side of the graph of . (iv) If neither nor are differentiable at then the function cannot be differentiable at . 
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Hint 

The tangent line of at the point is . 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The correct choice is (iii), because at the inflection point the function changes its concavity, so the tangent line moves from being above the graph to be below the graph or vice versa, so it must cross the graph from one side to the other.
(ii) is not correct. The tangent line of at a point is . Assume that there is a tangent line which goes through . This means that there exists a point such that . However there's no such point because the maximum of is . This is a contradiction. We find the maximum in the following way. Let . Then, its derivative is , which vanishes at . Indeed, these are critical points. so that the maximum in is achieved either the critical points or the end points . Since , the maximum is . (iv) is not correct because we can take , , and ; Obviously, and are NOT differentiable at . but is a constant function and hence is differentiable at .
For this purpose, we calculate the derivative . Since changes the sign around , doesn't change its sign around . For example, in the case that change its sign from negative to positive around , then for sufficiently close with , and for sufficiently close with , . Therefore, we prove that is either increasing or decreasing around .
