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Question Number 198279 by universe last updated on 16/Oct/23

if f(x) is also differentiable on R such that f′(x) > f(x) ∀ x ∈ R and f(x_0 ) = 0 then prove that f(x) ≥ 0 ∀ x > x_0

$$\:\:\mathrm{if}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{also}\:\mathrm{differentiable}\:\mathrm{on}\:\mathbb{R}\:\mathrm{such}\:\mathrm{that} \\ $$$$\:\:\mathrm{f}'\left(\mathrm{x}\right)\:>\:\mathrm{f}\left(\mathrm{x}\right)\:\forall\:\mathrm{x}\:\in\:\mathbb{R}\:{and}\:\mathrm{f}\left(\mathrm{x}_{\mathrm{0}} \right)\:=\:\mathrm{0}\:\mathrm{then}\: \\ $$$$\:\:\mathrm{prove}\:\mathrm{that}\:\:\mathrm{f}\left(\mathrm{x}\right)\:\geqslant\:\mathrm{0}\:\forall\:\mathrm{x}\:>\:\mathrm{x}_{\mathrm{0}} \\ $$

Answered by witcher3 last updated on 16/Oct/23

f′(x)−f(x)>0....(1) (1)∗e^(−x) ⇔ f(x)′e^(−x) −f(x)e^(−x) >0 (f(x)e^(−x) )′>0 ⇒∫_x_0 ^x (f(x)e^(−x) )′dx>∫_x_0 ^x 0dx f(x)e^(−x) −f(x_0 )e_(=0) ^(−x_0 ) ≥0 f(x)e^(−x) ≥0 ∀x>0,⇒f(x)≥0,x→e^(−x) >0,∀x∈[x_0 ,∞[

$$\mathrm{f}'\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{x}\right)>\mathrm{0}....\left(\mathrm{1}\right) \\ $$$$\left(\mathrm{1}\right)\ast\mathrm{e}^{−\mathrm{x}} \Leftrightarrow \\ $$$$\mathrm{f}\left(\mathrm{x}\right)'\mathrm{e}^{−\mathrm{x}} −\mathrm{f}\left(\mathrm{x}\right)\mathrm{e}^{−\mathrm{x}} >\mathrm{0} \\ $$$$\left(\mathrm{f}\left(\mathrm{x}\right)\mathrm{e}^{−\mathrm{x}} \right)'>\mathrm{0} \\ $$$$\Rightarrow\int_{\mathrm{x}_{\mathrm{0}} } ^{\mathrm{x}} \left(\mathrm{f}\left(\mathrm{x}\right)\mathrm{e}^{−\mathrm{x}} \right)'\mathrm{dx}>\int_{\mathrm{x}_{\mathrm{0}} } ^{\mathrm{x}} \mathrm{0dx} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\mathrm{e}^{−\mathrm{x}} −\mathrm{f}\left(\mathrm{x}_{\mathrm{0}} \right)\mathrm{e}_{=\mathrm{0}} ^{−\mathrm{x}_{\mathrm{0}} } \geqslant\mathrm{0} \\ $$$$\mathrm{f}\left(\mathrm{x}\right)\mathrm{e}^{−\mathrm{x}} \geqslant\mathrm{0} \\ $$$$\forall\mathrm{x}>\mathrm{0},\Rightarrow\mathrm{f}\left(\mathrm{x}\right)\geqslant\mathrm{0},\mathrm{x}\rightarrow\mathrm{e}^{−\mathrm{x}} >\mathrm{0},\forall\mathrm{x}\in\left[\mathrm{x}_{\mathrm{0}} ,\infty\left[\right.\right. \\ $$$$ \\ $$$$ \\ $$

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