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f-x-2x-3-x-4-show-that-f-x-0-has-roots-between-1-and-2-

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Question Number 66149 by Rio Michael last updated on 09/Aug/19
f(x) =2x^3 −x−4 show that f(x) =0 has roots between 1 and 2
f(x)=2x3−x−4showthatf(x)=0hasrootsbetween1and2
Answered by MJS last updated on 09/Aug/19
f(1)=−3<0 f(2)=10>0 and f(x) is continuous for x∈R ⇒ there nust be at least one zero between x=1 and x=2
f(1)=−3<0f(2)=10>0andf(x)iscontinuousforx∈R⇒therenustbeatleastonezerobetweenx=1andx=2
Answered by mr W last updated on 09/Aug/19
f′(x)=6x^2 −1=0 ⇒x=±(1/( (√6))) i.e. for x>(1/( (√6))) we have f′(x)>0 ⇒strictly increasing f(1)=2−1−4=−3<0 f(2)=16−2−4=10>0 ⇒f(x)=0 has one and only one root between 1 and 2.
f′(x)=6x2−1=0⇒x=±16i.e.forx>16wehavef′(x)>0⇒strictlyincreasingf(1)=2−1−4=−3<0f(2)=16−2−4=10>0⇒f(x)=0hasoneandonlyonerootbetween1and2.

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