### Home > APCALC > Chapter 5 > Lesson 5.3.2 > Problem 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 ✎

.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.has only one critical point (at) and. ifandif.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*).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.*h*has a global minimum at, and both the first and second derivatives are zero there. Nevertheless, ifandif.Refer to hint in part (b), the 1st Derivative Test.