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The range of the function g(x) = 2/(2 + 4x + 3x^2) can be written as an interval (a,b]. What is a+b?

 Jun 23, 2022
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We have a lower power polynomial over a higher power polynomial

 

This produces a  horizontal asymptote of     y =  0

 

This  is the bottom of the range

 

To discover the max use some Calculus

 

Take the first derivative of the function and set =  0

 

0 - 2 * (4 + 6x)

____________  =   0

(2 + 4x + 3x^2)

 

This simplifies to

 

4 + 6x =  0

 

4 =  -6x

 

x =  -4/6 =  -2/3

 

The max occurs at  x  = -2/3

 

The max is           2 /  ( 2 + 4(-2/3) + 3(-2/3)^2 )  =    2 / [ 2 -8/3 + 4/3 ] =  2 / [  2/3] =  3

 

 

The range is   (0, 3 ]

 

a + b =   0 + 3  =  3

 

 

cool cool cool

 Jun 23, 2022
edited by CPhill  Jun 23, 2022

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