Increasing and Decreasing Functions A function is increasing on a particular interval if for any, then Ie: As x increases, A function is decreasing on a particular interval if for any, then Ie: As x increases, For a continuous and differentiable function, f(x), the y-values are increasing for all x-values when and the y-values are decreasing for all x-values when Eg.1: Find the intervals of increase and decrease for the function, f (x) = 2x3 -3x2 -12x Interval Value of f '(x) Slope of Tangent Y-values are Eg.2: Find the intervals of increase and decrease for the function, f (x) = x -3x3 Eg.3: Consider the graph of f '(x). Sketch a possible graph of f(x). 2- V x 0 Solution: When the derivative, f '(x) is positive (above the x-axis), the graph of f(x) will be_ When the derivative is negative (below the x-axis), the graph of f(x) is_ The points on f '(x) = 0 (ie: _) are the critical points or _ on the graph of f(x). In this example, f '(x) = 0 @ x = therefore these are the turning points on the graph of f(x). Interval Value of f '(x) f(x) will be Therefore one possible graph of f(x) is