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Let f(x)=5-|x-2| and g(x)=|x+1|, x in R....

Let `f(x)=5-|x-2| and g(x)=|x+1|, x in R`. If f(x)n artains maximum value at `alpha` ang g(x) attains minimum value of `beta`, then `lim_(xto-alpha beta) ((x-1)(x^(2)-5x+6))/(x^(2)-6x+8)` is equal to

A

`1//2`

B

`-3//2`

C

`-1//2`

D

`3//2`

Text Solution

Verified by Experts

The correct Answer is:
A
Given function are f(x)=-5-|x-2|
and g(x) |x+1|, where `x in R`.
Clearly, maximum of f(x) occurred at `x=2, so alpha, 2`. And minimum of g(x) occurred at `x=- 1, so beta=-1`
`rArr alpha beta=-2`
Now, `underset(x to- alpha beta) lim ((x-1)(x^(2)-5x+6))/(x^(2)-6x+8)`
`underset(x to-2) lim ((x-1)(x-3)(x-2))/((x-4)(x-2))" " [ :. alpha beta =-2]`
`underset(x to-2) lim ((x-1)(x-3))/((x-4))=((2-1)(2-3))/((2-4))=(1xx(-1))/((-2))=(1)/(2)`
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