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.

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

   Hint (a):

   The 2nd Derivative Test states that if:
   If $f^\prime(a) = 0$ AND $f^{\prime\prime}(a) > 0$, then $x = a$ is the location of a local minimum on $f(x)$.
   If $f^\prime(a) = 0$ AND $f^{\prime\prime}(a) < 0$, then $x = a$ is the location of a local maximum on $f(x)$.
   But if $f^\prime(a) = 0$ AND $f^{\prime\prime}(a) = 0$, then the test is inconclusive...$f(a)$ might be a max, min or inflection point.

2. $g(x)$ 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 $x > 4$.

   Hint (b):

   The 1st Derivative Test states that if:
   If $f^\prime(a) = 0$ AND $f^\prime(x)$ changes from negative to positive values at $x = a$, then $x = a$ is the location of a local minimum on $f(x)$.
   If $f^\prime(a) = 0$ AND $f^\prime(x)$ changes from positive to negative values at $x = a$, then $x = a$ is the location of a local maximum on $f(x)$.
   But if $f^\prime(a) = 0$ AND $f^\prime(x)$ 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.

   Hint (c):

   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: $g^\prime(4) =$ DNE but $g^\prime(x)$ changes from negative to positive at $x = 4$.

4. $h(x)$ has a global minimum at $(0, 3)$, but the first and second derivatives are both zero there. Nevertheless, $h^\prime(x) > 0$ if $x > 0$ and $h^\prime(x) < 0$ if $x < 0$.

   Hint (d):

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