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5-123.

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.

  1. .

    The 2nd Derivative Test states that if:
    If AND , then is the location of a local minimum on .
    If AND , then is the location of a local maximum on .
    But if AND , then the test is inconclusive... might be a max, min or inflection point.

  2. has only one critical point (at ) and . if and if .

    The 1st Derivative Test states that if:
    If AND changes from negative to positive values at , then is the location of a local minimum on .
    If AND changes from positive to negative values at , then is the location of a local maximum on .
    But if AND does not change signs at , then is the location of an inflection point on .

  3. 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 .

  4. 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.