IIT JEE , AIEEE Forum - Mathematics , Algebra

beethoven

1.find the least valueof (x²-x+3)/(x-2)

2.if x=3+3^2/3+3^1/3,find the value of x³-9x²+18x.

nadeemoidu

1) 9
2) 12
If its correct i'll explain

beethoven

the answer for the second one is correct,but for the first one it is 1.
now plz explain me the second one

raja987654321

buddy i ve a serious doubt abt ur first prob x²-x+3/x-2 = x+1+5/x-2 , so the least value the expression can have can never be 1 rather it tends to -(infinity)

pls correct me if i m wrong

rishabh29288

yes the answer for first cannot be 1.........it is equivalent to x + 5/x-2 +1

nadeemoidu

x³ - 9x²+18x=x(x-3)(x-6)

=(3+ 3 ^2/3 + 3 ^1/3) ( 3 ^2/3 + 3 ^1/3) ( 3^2/3 + 3^1/3 - 3)

= ( 3 ^2/3 + 3 ^1/3) [( 3 ^2/3 + 3 ^1/3)² -3 ²]

= ( 3 ^2/3 + 3 ^1/3)[ 3 ^4/3 + 3 ^2/3+2*3 -9]

= ( 3 ^2/3 + 3 ^1/3)[ 3 ^4/3 + 3 ^2/3-3 ]
Now just expand and you get the answer

sggs

yes , the answer to the first can only be -(infinity).One another explanation is :

Let the given expression X² - X+ 3/X-2 = Y

then , X² -X + 3= YX - 2Y

X² -(Y+1)X +(2Y+3)=0

Now,for X to have a real value,

(b*b)-4ac>=0

Therefore,

(Y+1)² - 4(2Y+3)>=0

so the final expression comes out to be:

Y² - 6Y-11>=0

So for this inquality to hold good,

Y

( - infinity , 3 - ^{[2 ]}

5)

(3 + ^{[2 ]}

5,infinity)

So, yo can see that the least value of Y is -(infinity).

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