Using the graph of the slope function $f^\prime(x)$ below, determine where the following situations occur for $f(x)$:

1. Relative minima and maxima.

   Hint 1(a):

   Relative minima and maxima can exist when $f^\prime(x) = 0$. $x = a$, $x = d$ and $x = f.$ But which indicates a max and which indicates a min?

   Hint 2(a):

   To distinguish between a relative minimum and a relative maximum, use the 1st or 2nd derivative test.

   Hint 3(a):

   Refer to the hints in problem 5-42 for more guidance.

2. Intervals at which $f(x)$ is increasing.

   Hint (b):

   $f(x)$ increases when the slope is positive... and we are looking at graph of the slopes.

3. Inflection points.

   Hint (c):

   Inflection points can happen where $f^{\prime\prime}(x) = 0$... and we are looking at the graph of $f^\prime(x)$.

4. Intervals at which $f(x)$ is concave up and concave down.

   Hint (d):

   $f(x)$ is concave up when $f^{\prime\prime}(x) > 0$... and we are looking at a graph of $f^\prime(x)$.