MATH110 April 2019
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Question 01 (i)
Suppose is differentiable everywhere and assume it has a local maximum at the point . Does the function have a local extreme value at ? Justify your answer.
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The new function is differentiable at also. Therefore if it has a local max, then its derivative must be zero.
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Since is differentiable everywhere, . Note also that since is a point. The derivative of is
Evaluated at we have
Here, we know that because we are told that , and we also know that has a local maximum at , which means that .
Since the derivative is not zero, does not have a local extreme at .
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