## 5.3 First Derivative and Function Characteristics

#### Increasing and Decreasing Function

- f(x) is strictly increasing in an interval (a,b) if f^{\prime}(x)>0 for all values of x in that interval.

In this case, the tangent to any points in the interval has a positive gradient. - Similarly, f(x) is strictly decreasing in an interval if f^{\prime}(x)<0.

The tangent has a negative gradient. - If f(x) is said to be strictly increasing (or decreasing) without any interval specified, then f^{\prime}(x)\geq0 (or f^{\prime}(x)\leq0)for ANY values of x (or precisely the domain of f(x)).

Note: Notice the signs here are \geq, \leq instead of >, <.