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How you show with calculus method inequality x^2/x +1<=-4 if x<=-2?

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econominty | Student, Undergraduate | (Level 1) Honors

Posted May 17, 2013 at 1:54 PM via web

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How you show with calculus method inequality x^2/x +1<=-4 if x<=-2?

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crmhaske | College Teacher | (Level 3) Associate Educator

Posted May 25, 2013 at 3:10 PM (Answer #2)

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`f(x)=x^2/(x+1)lt=-4`

First we will rearrange the function to bring all terms to the same side:

`f(x)=x^2lt=-4(x+1)`

`f(x)=x^2+4x+4lt=0`

Using the calculus method, we must take the derivative of the function and show that f'(x) is decreasing for x<-2 and that x=-2 is a root of the function for f(x)<=0 for x<=-2:

`f'(x)=2x+4lt0`

`xlt-2`

Therefore, the function is decreasing for x<-2, and as such, x^2+4x+4<0 when x<-2.  As x=-2 is a root of the function, x^2+4x+4<=0 when x<=-2 is also true.

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