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A function is said to be increasing on an interval if, for any two points in the interval, the function value at the later point is greater than the function value at the earlier point. In other words, as the input (x) increases, the output (f(x)) also increases on that interval. Similarly, a function is said to be decreasing on an interval if the function value decreases as the input increases.

The derivative of a function, usually denoted by f'(x) or \text{d}f/\text{d}x, is a measure of how the function is changing at each point. In other words, it represents the rate of change or the slope of the tangent line to the graph of the function at a particular point. The first derivative tells us whether the function is increasing or decreasing, and by how much.

The sign of the first derivative (positive or negative) tells us whether the function is increasing or decreasing at a given point. If f'(x) > 0, then the function is increasing at x, because its rate of change is positive. If f'(x) < 0, then the function is decreasing at x, because its rate of change is negative. If f'(x) = 0, the function could potentially have a local maximum, local minimum, or a point of inflection.

Consider the function f(x) = x^3 - 3x^2 + 2x. First, we find the first derivative f'(x): f'(x) = 3x^2 - 6x + 2 Now, let's analyze the sign of f'(x) to determine whether the function is increasing or decreasing: 1. f'(x) > 0: 3x^2 - 6x + 2 > 0 This inequality holds true for intervals (0, 1) and (2, +∞). Therefore, the function f(x) is increasing on these intervals. 2. f'(x) < 0: 3x^2 - 6x + 2 < 0 This inequality holds true for the interval (1, 2). Therefore, the function f(x) is decreasing on this interval. To sum up, by examining the sign of the first derivative, we can determine where the function is increasing and decreasing. The function f(x) = x^3 - 3x^2 + 2x is increasing on intervals (0, 1) and (2, +∞) and decreasing on the interval (1, 2).