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if-x-R-prove-that-x-6-x-1-gt-0-




Question Number 151475 by mathdanisur last updated on 21/Aug/21
if  x∈R  prove that:  x^6  - x + 1 > 0
ifxRprovethat:x6x+1>0
Answered by dumitrel last updated on 21/Aug/21
I. if x≤0⇏−x≥0; x^6 +1>0⇒  x^6 −x+1>0  II. if x>0⇒x^6 +1=x^6 +(1/5)+(1/5)+(1/5)+(1/5)+(1/5)≥^(am−gm)   ≥((6x)/( (5^5 )^(1/6) ))>((6x)/5)>x⇒x^6 −x+1>0
I.ifx0x0;x6+1>0x6x+1>0II.ifx>0x6+1=x6+15+15+15+15+15amgm6x556>6x5>xx6x+1>0
Commented by mathdanisur last updated on 21/Aug/21
Nice Ser, Thank You
NiceSer,ThankYou
Answered by MJS_new last updated on 21/Aug/21
f(x)=x^6 −x+1  f ′(x)=6x^5 −1  f ′′(x)=30x^4   f ′(x)=0 ⇒ x=(1/6^(1/5) ); f((1/6^(1/5) ))≈.4>0  f ′′(x)>0∀x∈R ⇒  ⇒ absolute minimum >0
f(x)=x6x+1f(x)=6x51f(x)=30x4f(x)=0x=161/5;f(161/5).4>0f(x)>0xRabsoluteminimum>0

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