### Home > CALC > Chapter 5 > Lesson 5.3.2 > Problem 5-123

Sketch an example of a graph with the given characteristics. Assume the graph is continuous and differentiable everywhere unless you are told otherwise. Comment on local or global maxima and minima. Find a suitable function for as many as you can.

,, and.The 2nd Derivative Test states that if:

IfAND, thenis the location of a local minimum on.

IfAND, thenis the location of a local maximum on.

But ifAND, then the test is inconclusive...might be a max, min or inflection point.has only one critical point (at) and.ifandif.The 1st Derivative Test states that if:

IfANDchanges from negative to positive values at, thenis the location of a local minimum on.

IfANDchanges from positive to negative values at, thenis the location of a local maximum on.

But ifANDdoes not change signs at, thenis 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 butchanges from negative to positive at.has a global minimum at, but the first and second derivatives are both zero there. Nevertheless, ifandif.Refer to hint in part (b), the 1st Derivative Test.