### 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. Homework Help ✎

*f*′(0) =*f*′(4) = 0 ,*f*″(0) > 0 , and*f*″(4) < 0.*g*(*x*) has only one critical point (at*x*= 4) and*g*′(4) = 0.*g*′(*x*) < 0 if*x*< 4 and*g*′(*x*) > 0 if*x*> 4.Same as part (b), but

*g*′(4) is not defined.*h*(*x*) has a global minimum at (0, 3), but the first and second derivatives are both zero there. Nevertheless,*h*′(*x*) > 0 if*x*> 0 and*h*′(*x*) < 0 if*x*< 0.

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.

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*).

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.

Refer to hint in part (b), the 1st Derivative Test.