Use the graph of the given function to find any relative maxima and relative minima.

f(x) = x^3 - 3x^2 + 1

You will have to draw your own graph. The first derivative tells you that will be relative maxima or minima at x=0 and x=2. x=0 is a relative maximum and x=2 is a relative minimum

To find the relative maxima and minima of a function using its graph, we need to look for points where the graph changes from increasing to decreasing or from decreasing to increasing.

1. Locate the critical points: These are the points where the derivative of the function is zero or undefined. To find them, we need to find the derivative of the function f(x).

f'(x) = 3x^2 - 6x

Setting f'(x) = 0 and solving for x:
3x^2 - 6x = 0
x(3x - 6) = 0

From this equation, we find two critical points: x = 0 and x = 2.

2. Determine the intervals: We need to divide the x-axis into intervals using the critical points as boundaries. In this case, we have three intervals: (-∞, 0), (0, 2), and (2, ∞).

3. Check the sign changes: For each interval, we need to determine the sign of the derivative f'(x). We can choose any test point within each interval, plug it into f'(x), and observe if the result is positive or negative.

For the interval (-∞, 0), let's choose x = -1. Plugging this into f'(x),
f'(-1) = 3(-1)^2 - 6(-1)
f'(-1) = 3 + 6
f'(-1) = 9 (positive)

For the interval (0, 2), let's choose x = 1. Plugging this into f'(x),
f'(1) = 3(1)^2 - 6(1)
f'(1) = 3 - 6
f'(1) = -3 (negative)

For the interval (2, ∞), let's choose x = 3. Plugging this into f'(x),
f'(3) = 3(3)^2 - 6(3)
f'(3) = 27 - 18
f'(3) = 9 (positive)

4. Identify the extrema: By analyzing the sign changes, we can determine the behavior of the function in each interval.

- From (-∞, 0), the function is increasing (positive derivative)
- At x = 0 (critical point), the function changes from increasing to decreasing, implying a relative maximum.
- From (0, 2), the function is decreasing (negative derivative)
- At x = 2 (critical point), the function changes from decreasing to increasing, implying a relative minimum.
- From (2, ∞), the function is increasing (positive derivative)

So, we have a relative maximum at x = 0 and a relative minimum at x = 2.