Differentiable function proving help

[orir]

\(\displaystyle f \) is a differentiable function at \(\displaystyle [a,\infty)\).
i need to prove that:
if there is a constant \(\displaystyle m>0\) which maintains that \(\displaystyle f'(x)\geq m \) to every \(\displaystyle x\in[a,\infty),\)
so\(\displaystyle lim_{x\rightarrow\infty}f(x)=\infty \)... thx!!

 

[Deleted member 4993]

\(\displaystyle f \) is a differentiable function at \(\displaystyle [a,\infty)\).
i need to prove that:
if there is a constant \(\displaystyle m>0\) which maintains that \(\displaystyle f'(x)\geq m \) to every \(\displaystyle x\in[a,\infty),\)
so
\(\displaystyle lim_{x\rightarrow\infty}f(x)=\infty \)... thx!!

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[DrPhil]

\(\displaystyle f \) is a differentiable function at \(\displaystyle [a,\infty)\).
i need to prove that:
if there is a constant \(\displaystyle m>0\) which maintains that \(\displaystyle f'(x)\geq m \) to every \(\displaystyle x\in[a,\infty),\)
so\(\displaystyle lim_{x\rightarrow\infty}f(x)=\infty \)... thx!!

Bare bones approach .. please refer back to replies by daon2 and by JeffM to your previous question for more rigorous proof.

1. If there is no x such that f'(x) = 0, there is no relative maximum.
2. If no relative maximum, then the absolute maximum must be at an endpoint of the domain.
3. Since f'(x) is everywhere positive, the max is at the upper limit.
4. But there is no upper limit: no matter how large x is, since f'(x)>0 there is always an x for which f(x) is larger.

I am still interested in your use of the word "maintain" in English. Is that a translation of "mantener"? If so, I would like to be more familiar with mathematical vocabulary in Spanish. The way such a theorem would usually be expressed in English is
"If there exists a constant \(\displaystyle m>0\) such that \(\displaystyle f'(x)\geq m \) for every \(\displaystyle x\in[a,\infty),\) then \(\displaystyle lim_{x\rightarrow\infty}f(x)=\infty \)."