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if-f-x-is-also-differentiable-on-R-such-that-f-x-gt-f-x-x-R-and-f-x-0-0-then-prove-that-f-x-0-x-gt-x-0-

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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
iff(x)isalsodifferentiableonRsuchthatf′(x)>f(x)∀x∈Randf(x0)=0thenprovethatf(x)⩾0∀x>x0
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 ,∞[
f′(x)−f(x)>0….(1)(1)∗e−x⇔f(x)′e−x−f(x)e−x>0(f(x)e−x)′>0⇒∫x0x(f(x)e−x)′dx>∫x0x0dxf(x)e−x−f(x0)e=0−x0⩾0f(x)e−x⩾0∀x>0,⇒f(x)⩾0,x→e−x>0,∀x∈[x0,∞[

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