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Differentiable function

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#1 h Feb 18, 2025, 5:28 PM

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Let $f:\left [ 0,+\infty \right ] \rightarrow \mathbb{R}$ be a twice differentiable function satisfying:
$\lim_{x \to +\infty} (f''(x) - f'(x)) = b \neq 0.$ Prove that there exists $x_0$ such that $f(x) \neq 0, \quad \forall x \in (x_0, +\infty).$

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#2 h Feb 18, 2025, 11:37 PM • 1 Y

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Mod $f\rightarrow -f$ we may assume $b>0$ . Set $g(x):=f'(x)e^{-x}$ then $g'=(f''-f')e^{-x}$ hence $g'(x)>0$ for $x\ge A$ . If $g(t)>0$ for some $t\ge A$ then we set $\gamma:=g(t)$ and we have $g'(x)\ge \gamma >0$ for $x\ge t$ . Hence $f'(x)\ge \gamma e^x$ for $x\ge A$ . Then $f'(\infty)=\infty$ and the result is clear.
From now on we assume $g(t)\le 0$ for $t\ge A$ . Then $f$ decreases on $[A,\infty)$ .
If the result is not true, then $f=0$ on some interval $[B,\infty)$ . This contradicts the hypothesis, since on $(B,\infty)$ we have $f'(x)=f''(x)=0$ .

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