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[h]SL 5.2 Increasing and decreasing functions

[q] Increasing/Decreasing Functions

Direction of Slope

As you may remember from studying linear functions, any increasing function [→] has a positive slope, and any decreasing function [↓] has a negative slope, and a flat/horizontal line [→] has slope = 0.

[q]

When we had linear functions, those were constant slopes, but the same theory applies to gradient functions, and due to this, we can have curves that switch between increasing and decreasing.

When f'(x) > 0, the graph is increasing.

When f'(x) < 0, the graph is decreasing.

When f'(x) = 0, the graph is flat.

[a]

We can use this to assess whether functions are increasing/decreasing or which sections of the graph are increasing/decreasing.

Finding when f'(x) = 0 can tell you when the graph is flat, which can actually tell you where a maximum or minimum is located.

Examples:

E.G.: Is f(x) = e²ˣ an increasing or decreasing function, or both?

The derivative of f(x) = e²ˣ is f'(x) = 2e²ˣ, so we can check whether 2e²ˣ is positive, negative, or both.

Whatever x we choose (-ve, 0, +ve, etc.), it always gives a positive value.

This means it is an increasing function, which we see in its graph.

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