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Find the critical numbers and stationary points of each of the following functions.$\;f(x)=\large\frac{x+1}{x^{2}+x+1}$

Note: This is part 4th of a 6 part question, split as 6 separate questions here.

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Toolbox:
  • A ciritical number of a function f is a number c in the domain of f such that either $f'(c)=0$ or $f'(c)$ does not exist.
  • Stationary points correspond to critical numbers for which $f'(c)=0$
$f(x)=\large\frac{x+1}{x^2+x-1}$
Step 1:
$f'(x)= \large\frac{(x^2+x+1)-(x+1)(2x+1)}{(x^2+x+1)^2}$
$\qquad= \large\frac{x^2+x+1-(2x^2+3x+1)}{(x^2+x+1)^2}$
$\qquad= \large\frac{-x^2-2x}{(x^2+x+1)^2}$
$\qquad= \large\frac{-x(x+2)}{(x^2+x+1)^2}$
$f'(x)=0=>x(x+2)=0=>x=0,x=-2$ (critical numbers)
Step 2:
When $x=0, f(0)=1$
When $x=-2, f(-2) =\large\frac{-1}{4-2+1}=-\frac{1}{3}$
The critical points are $(0,1),(-2,-\large\frac{1}{3})$

 

answered Jul 31, 2013 by meena.p
 

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