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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. Homework Help ✎

  1. .

    The 2nd Derivative Test states that if:
    If f ′(a) = 0 AND f ′′(a) > 0, then x = a is the location of a local minimum on f(x).
    If f ′(a) = 0 AND f ′′(a) < 0, then x = a is the location of a local maximum on f(x).
    But if f ′(a) = 0 AND f ′′(a) = 0, then the test is inconclusive...f(a) 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 f ′(a) = 0 AND f ′(x) changes from negative to positive values at x = a, then x = a is the location of a local minimum on f(x).
    If f ′(a) = 0 AND f ′(x) changes from positive to negative values at x = a, then x = a is the location of a local maximum on f(x).
    But if f ′(a) = 0 AND f ′(x) does not change signs at x = a, then x = a is the location of an inflection point on f(x).

  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: g ′(4) = DNE but g ′(x) changes from negative to positive at x = 4.

  4. h 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.