If $g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous derivable function and $|g'(x)|\le |g(x)|$, can we find the function?

Asked Apr 25 '23 at 08:32

Active Apr 25 '23 at 08:32

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The exact wording of the question is given below

If $g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous derivable function such that g(0)=0 and $|g'(x)|\le |g(x)|\forall x\in\mathbb{R}$, then the value of $g(1)+g'(1)+g(2)+g'(2)$ cannot be:
(A) 1
(B) -1
(C) 3
(D) 7

All the options are correct. There must be some way to find the set of functions of $g(x)$. In fact, if mod function was not there, i.e, $g'(x)\le g(x)\forall x\in\mathbb{R}$, then it was a simple linear differential equation. I'm struggling to remove the mod and find the function.

How can we analyze the given inequality and make the correct conclusions?

asked Apr 25 '23 at 08:32

Pravimish

Mean value theorem – Charlie Frohman Apr 25 '23 at 08:33
2

Does this answer your question? $f(0)=0$ with $\left| f'(x)\right| \leq \left| f(x)\right|$ for $x\in\mathbb{R}$. Show that $f$ is identically zero. – Apass.Jack Apr 25 '23 at 08:59
1

Here are several more proofs. – Apass.Jack Apr 25 '23 at 09:03

If $g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous derivable function and $|g'(x)|\le |g(x)|$, can we find the function?

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