Note that what is shown is the graph of the

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Note that what is shown is the graph of the derivative , not the graph of !! Critical points at and . Neither local max nor min at ; neither local max nor min at ; local maximum at . Critical points at and . Local minimum at ; local maximum at . Critical points at , and . Neither local maximum nor minimum at ; local maximum at . Critical points at , and . Local minimum at ; local maximum at ; neither local maximum or minimum at . Correct! Correct! Critical points at and . Local maximum at ; neither local min nor local max at ; local minimum at .
The given graph is the graph of the derivative . We see that at and . So these are the three critical points. We use the first derivative test to tell where has a local minimum, a local maximum, or neither at each point. At , changes from negative to positive, so a local minimum at At , changes from positive to negative so has a local maximum at At , does not change sign (it is negative on either side of ), so has neither a local max nor a local min at . has . .
2.5 / 2.5 pts Question 8 Suppose is a continuous function that has critical points at and at such that and . The second derivative of given as . Use the second derivative test and choose the correct statement regarding local extrema at the given critical points. is .
has a local min at ; the second derivative test tells us nothing about what happens at . Correct! Correct! has a local max at ; has neither a local max nor a local min at . has a local min at and a local max at .
has a local max at ; the second derivative test tells us nothing about what happens at . If is positive at a critical point , then has a local minimum at . If is negative at a critical point , then has a local maximum at . If at a critical point , then the second derivative test tells us nothing about what happens at . , so the second derivative test tells us nothing about what happens at . , so has a local minimum at .

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