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Question

Let f(x) = ax2+bx+c,a>0 such that f(1x)=f(1+x) x ε R. Also given that f(x) = 0 has no real roots and b > O
Let α=4a2b+c,β=9a+3b+c,γ=9a3a+c. Then which of the following is correct?


Your Answer
A
β<α<γ
Your Answer
B
γ<α<β
Correct Answer
C
α<γ<β
Your Answer
D
α<β<γ

Solution

The correct option is C α<γ<β

f(1x)=f(1+x)f(x) is symmetric about x = -1
So vertex of parabola must be at x = –1 and it is concave up.
b2a=1b=2a........(i)α=f(2)=4a2b+cβ=f(3)=9a+3b+cγ=f(3)=9a3b+c
As f(x) has no real roots so values of f(x) increase as we move further away from x = –1 on both sides
Hence f(3)>f(3)>f(2)β>γ>α.
 

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