I learned that linear combinations of monotonically increasing or decreasing functions are either monotonically increasing or monotonically decreasing, But not monotonic. So are there any kind of combination of monotonic functions that create non monotonic functions like $ \sin (x) $. Or do we always need a non monotonic function to create another non monotonic function.
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$\begingroup$ Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. $\endgroup$– Community BotFeb 17 at 14:33
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$\begingroup$ $f(x) = x^3$ and $g(x) = -10x$ are monotonic functions, but $f+g$ is not monotonic, so your 1st affirmation is wrong. $\endgroup$– LourrranFeb 17 at 15:52
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1$\begingroup$ A linear combination of monotonically increasing functions with all coefficients positive is monotonically increasing. A linear combination of monotonically increasing functions with all coefficients negative is monotonically decreasing. A linear combination of monotonically decreasing functions with all coefficients positive is monotonically decreasing. A linear combination of monotonically decreasing functions with all coefficients negative is monotonically increasing. In other cases the results could be anything. $\endgroup$– Robert IsraelFeb 17 at 16:12
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