  ### Home > CALC > Chapter 5 > Lesson 5.3.2 > Problem5-123

5-123.
1. 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 ✎

1. f ′(0) = f ′(4) = 0 , f ″(0) > 0 , and f ″(4) < 0.

2. 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.

3. Same as part (b), but g ′(4) is not defined.

4. 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.