Q. 83.6( 15 Votes )

If f(x) = ax

Answer :

Given;


f(x) = ax2 + bx + c has no real zeroes, and a + b + c<0


Suppose a = – 1,


b = 1,


c = – 1


Then a + b + c = – 1,


b2 – 4ac = – 3


Therefore it is possible that c is less tha zero.


Suppose c = 0


Then b2 – 4ac = b2 ≥ 0


So,


f(x) has at least one zero.


Therefore c cannot equal zero.


Suppose c > 0.


It must also be true that b2 ≥0


Then,


b2 – 4ac < 0 only if a > 0.


Therefore,


a + b + c < 0.


– b > a + c > 0


b2 > (a + c)2


b2 > a2 + 2ac + c2


b2 – 4ac > (a – c)2 ≥ 0


As we know that the discriminant can’t be both greater than zero and less than zero,


So, C can’t be greater than zero.

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