### 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.

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 $f ′\left(a\right) = 0$ AND $f′′(a)>0$, then $x = a$ is the location of a local minimum on $f\left(x\right)$.
If $f ′\left(a\right) = 0$ AND $f′′(a)<0$, then $x = a$ is the location of a local maximum on $f\left(x\right)$.
But if $f ′\left(a\right) = 0$ AND $f ′′\left(a\right) = 0$, then the test is inconclusive...$f\left(a\right)$ 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 $f ′\left(a\right) = 0$ AND $f ′\left(x\right)$ changes from negative to positive values at $x = a$, then $x = a$ is the location of a local minimum on $f\left(x\right)$.
If $f ′\left(a\right) = 0$ AND $f ′\left(x\right)$ changes from positive to negative values at $x = a$, then $x = a$ is the location of a local maximum on $f\left(x\right)$.
But if $f ′\left(a\right) = 0$ AND $f ′\left(x\right)$ does not change signs at $x = a$, then $x = a$ is the location of an inflection point on $f\left(x\right)$.

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: $g ′\left(4\right) =$ DNE but $g ′\left(x\right)$ 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.