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How do you prove that a function for example is increasing without taking the derivative of it? What is that called? Monotonicity or something? A video explaining the concept would be nice but all I have found uses derivatives which I am not allowed to use.

For example prove that the function f(x) is increasing:

f(x) = 12x - 4100 if x < -2, f(x) = (x-14)^3 if x >= -2

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  • $\begingroup$ In some cases it's trivial. E.g. $x^3$ increases as $x$ increases, and $1/x$ decreases if $x$ increases without changing signs. Likewise $2^x$ increases as $x$ increases. And $\sin x$ increases as $x$ increases between $\pm\pi/2. \qquad$ $\endgroup$ Commented Nov 12, 2017 at 18:41

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By the definition of increasing function: You take arbitrary $x_{1},x_{2}\in D(f)$ such that $x_{1}\leq x_{2}$ and show that $f(x_{1})\leq f(x_{2})$. Similar for decreasing. Your example is piecewise function. Try to split it on three problems. Show monotonicity for single pieces. Then try to take $x_{1}\leq-2 \leq x_{2}$, if necessary.

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Yep, monotonicity. It means that the function is only increasing or decreasing on whatever interval your talking about.

For $x< -2$ we'll use the subscript $l$ to denote any $x$ selected that's to the left (less than) any other selected $x$ that's greater than $x_l$, call that one $x_r$

\begin{align*} x_l &< x_r \\ 12\cdot x_l &< 12 \cdot x_r \\ 12\cdot x_l - 4100 &< 12 \cdot x_r - 4100 \end{align*}

So you should be able to do the other one.

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