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Finding The Set of Values of x For Which a Graph is Increasing or Decreasing

The general gradient of a graph is given by the gradient function

To find the turning points of a graph we solve the equation

To find the set of values for which the graph is going up or increasing we solveIn general we will get an interval or set of intervals.

To find the set of values for which the graph is going down or decreasing we solve Again, in general we will get an interval or set of intervals.

Example: Find the set of values offor whichis increasing.

We solve for

Example: Find the set of values offor whichis decreasing.

We solve for

To see why this is the case we can sketch the graph of

We are looking for those values of x on the graph where the graph is less than 0 ie