Find all existing vertical and horizontal asymptotes,maximum and minimum and point of inflection of the graph y=(x+1)/(x^2+x+1)

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- Apr 29th 2006, 02:38 AMbobby77please help -very urgent...
Find all existing vertical and horizontal asymptotes,maximum and minimum and point of inflection of the graph y=(x+1)/(x^2+x+1)

- Apr 29th 2006, 03:32 AMTD!
HA -> check the limit for x going to +/- infinity

VA -> find the poles (zeroes of the denominator, but not of nominator) and check the limit approaching that point from both sides

min/max -> solve f'(x) = 0 and use the f" to see whether it's a min or max

inflection -> solve f"(x) = 0 and see if f" changes sign in those points - Apr 29th 2006, 11:42 AMbobby77please check my answer
please check my answer

- Apr 29th 2006, 05:50 PMAradesh
we have

vertical asymptotes occur when the denominator is zero

when

the discriminant

thus there are no real roots, hence no vertical asymptotes.

so as then from above

and as then from below

considering turning points.

from the information we know there must be two turning points, a minimum less than zero, and a maximum, greater than zero. also there cannot be more than this else it would mean there could be a horizontal line intersecting the curve in more than two places, which defies the fact our function is of second degree.

if we look at the horizontal lines

we know that the curve does not exist for values of c such that

has no real solutions.

so using this we can find the range of the curve for all values of c which there are real solutions, and thus find the turning points.

call this (1)

so this has real values when

so at y=-1/3 we have a minimum. and y = 1 we have a maximum. for which we can find corresponding x values, using (1) and c = -1/3

so

so there is a minimum at

and for the maximum y=1 we have from (1)

so x = 0 giving coordinate of maximum

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