(i) The domain of f(x) is all real numbers, since there are no restrictions on the input variable x.
(ii) We have:
f(x) = x^4 - 2x^3
f'(x) = 4x^3 - 6x^2
f''(x) = 12x^2 - 12x
(iii) To find the critical points, we need to solve the equation f'(x) = 0:
4x^3 - 6x^2 = 0
2x^2(2x - 3) = 0
x = 0 or x = 3/2
So the critical points are (0,0) and (3/2, -27/16).
(iv) To determine where f(x) is increasing or decreasing, we need to examine the sign of f'(x) on different intervals. We can make a sign chart for f'(x):
| x | -∞ | 0 | 3/2 | +∞ |
|---------|--------|-------|-------|--------|
| f'(x) | - | 0 | + | + |
From the sign chart, we see that f(x) is decreasing on the interval (-∞, 0) and increasing on the interval (0, 3/2) and (3/2, +∞).
(v) To find the relative extrema, we need to examine the sign of f'(x) around the critical points. We can make a table:
| x | 0- | 0+ | 3/2- | 3/2+ |
|---------|-------|-------|-------|-------|
| f'(x) | - | + | - | + |
| f(x) | 0 | 0 | -27/16| -27/16|
From this table, we see that f(x) has a relative minimum of -27/16 at x = 3/2, and no relative maximum or minimum at x = 0.
(vi) To find the intervals where f(x) is concave up or concave down, we need to examine the sign of f''(x) on different intervals.
We can make a sign chart for f''(x):
| x | -∞ | 0 | 1 | +∞ |
|---------|--------|-------|------|--------|
| f''(x) | + | - | + | + |
From the sign chart, we see that f(x) is concave down on the interval (-∞, 0) and concave up on the intervals (0, 3/2) and (3/2, +∞).
(vii) To find the points of inflection, we need to solve the equation f''(x) = 0:
12x^2 - 12x = 0
12x(x - 1) = 0
x = 0 or x = 1
So the points of inflection are (0,0) and (1, -1).
(viii) To sketch the graph of f(x), we can use the information we have gathered so far.
At x = 0, f(x) has a relative minimum of 0 and is concave down. At x = 3/2, f(x) has a relative minimum of -27/16 and is concave up. The point (1, -1) is a point of inflection.
Based on this information, we can sketch a graph of f(x) that looks like this:
```
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--------o------------o-------
0 3/2 x-axis
```
The graph is a "U" shape that opens upward, with a relative minimum at (3/2, -27/16) and a point of inflection at (1, -1).