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

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. $f ^\prime (0) = f ^{\prime }(4) = 0, f ^{\prime \prime}(0) > 0, \text{and}\ f ^{\prime \prime}(4) < 0$.

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.

2. $g$ has only one critical point (at $x = 4$) and $g^\prime(4) = 0$. $g^\prime(x) < 0$ if $x < 4$ and $g^\prime(x) > 0$ if $x > 4$.

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 $g^\prime(4)$ 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 $(0, 3)$, and both the first and second derivatives are zero there. Nevertheless, $h^\prime(x) > 0$ if $x > 0$ and $h^\prime(x) < 0$ if $x < 0$.

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