For each part below, sketch a graph with the given characteristics. Assume the function is continuous and differentiable everywhere unless you are told otherwise. Comment on local or global extrema. Write the equation of a possible function for as many parts as you can.
The 2nd Derivative Test states that if:
AND , then is the location of a local minimum on .
AND , then is the location of a local maximum on .
AND , then the test is inconclusive... might be a max, min or inflection point. has only one critical point (at ) and . if and if .
The 1st Derivative Test states that if:
AND changes from negative to positive values at , then is the location of a local minimum on .
AND changes from positive to negative values at , then is the location of a local maximum on .
AND does not change signs at , then is the location of an inflection point on .
Same as part (b), but
is not defined.
Functions can have an undefined slope (derivative) at a cusp, endpoint, jump, hole or vertical tangent.
Do any of those scenarios match the given one:
DNE but changes from negative to positive at . has a global minimum at , and both the first and second derivatives are zero there. Nevertheless, if and if .
Refer to hint in part (b), the 1st Derivative Test.