Increasing & Decreasing Functions
Recall that upward sloping straight lines have a positive gradient whereas downward sloping straight lines have a negative gradient. The same applies to curves. Gradients on a curve are always changing but an upward sloping curve has a positive gradient and a downward sloping curve has a negative gradient.
Recall the graph of . You will notice that for positive x, the graph has a positive gradient; for negative x the graph has a negative gradient; and for x=0 the gradient is also 0.
This can be seen from the gradient function . Find out more about differentiating. 2x is positive when x is positive, negative when x is negative and 0 when x is 0.
Example 1– Find the range of values of x for which the graph of has a positive gradient.
Differentiating y gives . This is positive when , i.e. when 2x is greater than 5. This gives the solution .
Example 2 – Explain why the gradient of is never negative.
You can see from the graph of x cubed that it never has a negative gradient but we show it using differentiation.
which is 3 lots of a square number. Irrespective of the value of x that is put in, this number will always be positive.