## 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 $y=x^2$. 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 $\frac{dy}{dx}=2x$. Find out more about differentiating. 2x is positive when x is positive, negative when x is negative and 0 when x is 0.

Example 1Find the range of values of x for which the graph of $y=x^2-5x+4$ has a positive gradient.

Differentiating y gives $\frac{dy}{dx}=2x-5$. This is positive when $2x-5\textgreater 0$, i.e. when 2x is greater than 5. This gives the solution $x\textgreater 2.5$.

Example 2 Explain why the gradient of $y=x^3$ 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.

$\frac{dy}{dx}=3x^2$ which is 3 lots of a square number. Irrespective of the value of x that is put in, this number will always be positive.