Science:Math Exam Resources/Courses/MATH110/April 2013/Question 01 (c)

MATH110 April 2013

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[hide] Question 01 (c)
Determine whether the following statement is true or false. If it is true, provide justification. If it is false, provide a counterexample. (c) If ƒ has a local minimum at 1, then ƒ'(1) = 0.

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[show] Hint
Recall the definition of a critical point. Is there more than just the criteria that its derivative is zero?

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[show] Solution
The statement is false. A minimum always occurs at a critical point - however, there are two possible reasons for a critical point: either $\displaystyle f'(x)=0$ or $\displaystyle f'(x)$ does not exist. In this second case, the derivative is not equal to zero, but there can still be a local minimum. For example, the derivative does not exist if there is a cusp (sharp corner), a vertical tangent or the function is discontinuous; an example of a cusp is given below: In this case there is a minimum at x = 1 but $\displaystyle f'(x)$ does not exist, so it can't be equal to zero. So we found a counter example and hence the statement is false.