1. SOLVE
| 2x + 1 | - | 2 x - 2 | = 5 ans:[2,∞)
2.Prove that x5+ax4+bx3+cx2+dx+e=0
cannot all have real roots if , 2a2<5b
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3 Answers
62first part is easy dear.. (If u dont feel it easy, it is a standard questions)
you need to find the different intervals..
here they are formed by 2x+1=0 and 2x-2=0
hence, x=-1/2 and x=1
So check at (- infinity , -1/2 ) (-1/2, 1) and (1, infintiy)
take 3 intervals.. u will get the solution :)
62Quesiton 2:
The conditions for all real roots is that the kth derivative will have all real roots!
Why does this statement work?
Even if u dont understand it try to apply this.. then we will see why this statement is aactually true..
applyitng this above result, u will get the answer :)
62See if there is a polynomial of nth degree.. to have n roots, it should cut the x axis n times...
so it must change sign n-1 times...
so its derivative must be 0 n-1 times
so u can keep proceeding this way.. then u will get that the kth derivative should have all real roots :)
now u need to apply this to the question here...
How will u do that ?
any first ideas?
