Checking if f(x)=x^3 is Monotonic: No Derivatives Needed

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SUMMARY

The function f(x) = x^3 is monotonically increasing for all real numbers. This conclusion is reached by applying the definition of monotonicity without the use of derivatives. Specifically, for any positive real number h, if the condition f(x+h) - f(x) ≥ 0 holds, then the function is confirmed to be monotonically increasing. This method effectively demonstrates the behavior of the cubic function across its entire domain.

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Yankel
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Hello,

I want to check if

f(x)=x^3 is monotonically increasing or monotonically decreasing or not monotonic at all.

How do I do that, without using derivatives yet ?

Thanks !
 
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You use the definition.
 
Let $h$ be some positive real number, then if:

$$f(x+h)-f(x)\ge0$$ the function is monotically increasing or if:

$$f(x+h)-f(x)\le0$$ the function is monotically decreasing.
 

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