# Differentiation: Increasing and Decreasing Functions

## Understanding Increasing and Decreasing Functions

• A function is deemed as increasing on an interval if the function values increase as the x-values move from left to right over that interval. Essentially, for any two points, if the x-value of the second point is larger, then its y-value will also be larger.
• A function is said to be decreasing on an interval if the function values decrease as the x-values move from left to right over that interval. In other words, for any two points on the graph during this interval, if the x-value of the second point is larger, its y-value will be smaller.
• A function can be both decreasing and increasing over different intervals.

## Differentiation and Increasing/Decreasing Functions

• Differentiation can be used to determine intervals where a function is increasing or decreasing.
• If the derivative of the function, f’(x), is positive over an interval, the function is increasing over that interval.
• If the derivative f’(x) is negative over an interval, the function is decreasing over that interval.
• It is important to remember that these rules apply to the intervals where the function is differentiable, ignoring the points at which f’(x) is undefined.

## Identifying Turning Points

• A turning point is a point at which the function changes from increasing to decreasing, or decreasing to increasing. These are also known as ‘local minimum’ and ‘local maximum’ points.
• By setting the derivative f’(x) equal to zero and solving for x, you can find potential turning points.
• By examining the sign of the derivative function to the left and right of these points, you can determine if the function really does change from increasing to decreasing (a local maximum) or decreasing to increasing (a local minimum) at these points.

## Applications in Real Life

• Understanding the concept of increasing and decreasing functions aids in interpreting and predicting real-world situations modelled by functions, such as population growth, economic trends, and physical phenomena.
• Skills in identifying intervals where a function increases or decreases have vital applications in many fields including physics, engineering, economics, and biology.