What are the critical values, if any, of $f(x)= (x+1) /(x^2 + x + 1)$ ?

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Answer 1 · 2016-11-10T04:58:34

The critical values will occur when the derivative is $0$ or undefined.

$f'(x) = (1(x^2+ x +1) - (x + 1)(2x + 1))/(x^2 + x + 1)^2$

$f'(x) = (x^2 + x + 1 - (2x^2 + 3x + 1))/(x^2 + x+ 1)^2$

$f'(x) = (-x^2 - 2x)/(x^2 + x + 1)^2$

Set the derivative to $0$ and solve. Also, find the vertical asymptotes (where the function is undefined).

$0 = (-x^2 - 2x)/(x^2 + x+ 1)^2$

$0 = -x^2 - 2x$

$0 = -x(x + 2)$

$x = 0 and -2$

For V.A:

$(x^2 + x + 1)^2 = 0$

$x^2 + x + 1 = 0$

$x = (-1 +- sqrt(1^2 - 4 xx 1 xx 1))/(2 xx 1)$

$x = (-1 +- sqrt(-3))/2$

$:.$ There are no vertical asymptotes.

Hence, the critical numbers are $x= 0$ and $x= -2$ .

Hopefully this helps!

What are the critical values, if any, of f(x)= (x+1) /(x^2 + x + 1)f(x)= (x+1) /(x^2 + x + 1)?