Decreasing Functions

The graph of a function is all we need to determine where it is decreasing.

Find the intervals on the x axis over which the polynomial function f(x) = x³ + 2x² - 3x - 6 is decreasing.

• Graph the function (see Graphing Functions)
  
  

• The parts of the graph above highlighted in blue are the parts of the graph that are decreasing (y coordinates are decreasing as the x coordinates increase). The interval on the x-axis where the function is decreasing is highlighted in red. This interval is the one we want to find.
  
  
• The left endpoint of this interval is the x-coordinate of the relative maximum value and its right endpoint is the x-coordinate of the relative minimum value. These are found (see Finding Relative Maximum Values and Finding Relative Minimum Values) to be -1.868516961 and .53518379667. So the interval is [-1.868516961, .53518379667].
  

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Over what intervals is the rational function f(x) = decreasing?

• Graph the function;
  
  

• The part of the graph above highlighted in blue is the part of the graph that is decreasing (y coordinates are decreasing as the x coordinates increase). The interval on the x-axis where the function is decreasing is highlighted in red. This is the interval we want to find.
  
  
• The red interval goes from minus infinity to the x-coordinate of the relative minimum point. We find this to be x= 0 (see Finding Relative Minimum Values).