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Qu: Let a 1 ,a 2 .........a n be greater than or equal to zero. P(x)= x n +a 1 x n-1 +.........+a n-1 x+1 has n real roots. find least value of P(2). options: a)3n b)2n c) 1 d) none

Qu: Let a1,a2.........an be greater than or equal to zero.  P(x)= xn+a1xn-1+.........+an-1x+1  has n real roots.  find least value of P(2).


options: a)3n  b)2n c) 1  d) none

Grade:

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
6 years ago
Ans:
Hello Student,
Please find answer to your question below

P(x) = x^n + a_{1}x^{n-1} + a_{2}x^{n-2} + a_{3}x^{n-3} + ..........+a_{n-1}x + a_{n}
Since a1, a2, a3…......anall are greater than or equal to zero.
P(2) = 2^n + a_{1}2^{n-1} + a_{2}2^{n-2} + a_{3}2^{n-3} + ..........+a_{n-1}2 + a_{n}
Since all the coefficients are greater than or equal to zero. No coeffiecient is negative.
So minimum of P(2) will be at when all the coefficients are zero.
[P(2)]_{min} = 2^n

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