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Answer:

If [tex]f(x_1)\leq f(x_2)[/tex] whenever [tex]x_1\leq x_2[/tex] f is increasing on I.

If [tex]f(x_1)\geq f(x_2)[/tex] whenever [tex]x_1\leq x_2[/tex] f is decreasing on I.

Step-by-step explanation:

These are definitions for real-valued functions f:I→R. To help you remember the definitions, you can interpret them in the following way:

When you choose any two numbers [tex]x_1\leq x_2[/tex] on I and compare their image under f, the following can happen.

  • [tex]f(x_1)\leq f(x_2)[/tex]. Because x2 is bigger than x1, you can think of f also becoming bigger, that is, f is increasing. The bigger the number x2, the bigger f becomes.
  • [tex]f(x_1)\geq f(x_2)[/tex]. The bigger the number x2, the smaller f becomes so f is "going down", that is, f is decreasing.

Note that this must hold for ALL choices of x1, x2. There exist many functions that are neither increasing nor decreasing, but usually some definition applies for continuous functions on a small enough interval I.

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