If f(x)=ax^2+bx+c has no real zeroes and a+b+c<0,then
(a)c=0 (b)c>0 (c)c<o (d)none of these

Asked by satapathy99 | 24th May, 2014, 06:54: PM

Expert Answer:

G i v e n space e q u a t i o n space i s space a x squared plus b x plus c equals 0 A l s o space g i v e n space t h a t space t h e space r o o t s space o f space t h e space a b o v e space e q u a t i o n space a r e space n o t space r e a l. H e n c e space t h e space d i s c r i m i n a n t space b squared minus 4 a c less than 0  C a s e space left parenthesis a right parenthesis : space I f space c equals 0 comma space t h e n space t h e space d s c r i m i n a n t space D equals b squared minus 4 a open parentheses 0 close parentheses less than 0 rightwards double arrow b squared less than 0 S q u a r e d space t e r m space c a n n o t space b e space n e g a t i v e. T h u s comma space c space c a n n o t space b e space e q u a l space t o space z e r o.  C a s e left parenthesis b right parenthesis : space A s s u m e space t h a t space c greater than 0 I n space t h e space d i s c r i m i n a n t comma space b squared space i s space p o s i t i v e. S o space t h e space d i s c r i m i n a n t space w i l l space b e space n e g a t i v e space o n l y space i f space a greater than 0 N o w space c o n s i d e r space t h e space g i v e n space i n e q u a l i t y space a plus b plus c less than 0 rightwards double arrow b less than minus a minus c rightwards double arrow minus b greater than left parenthesis a plus c right parenthesis rightwards double arrow b squared greater than left parenthesis a plus c right parenthesis squared rightwards double arrow b squared greater than a squared plus c squared plus 2 a c rightwards double arrow b squared minus 4 a c greater than a squared plus c squared minus 2 a c rightwards double arrow b squared minus 4 a c greater than left parenthesis a minus c right parenthesis squared greater or equal than 0
T h e space d i s c r i m i n a n t space c a n n o t space b e space l e s s space t h a n space z e r o space a n d space g r e a t e r space t h a n space z e r o space s i m u l tan e o u s l y. T h u s comma space a space c a n n o t space b e space g r e a t e r space t h a n space z e r o. rightwards double arrow c space c a n n o t space b e space g r e a t e r space t h a n space z e r o.  C a s e left parenthesis c right parenthesis : S o space t h e space o n l y space p o s s i b i l i t y space i s space c less than 0 T h u s comma space t h e space c o r r e c t space o p t i o n space i s space c.

Answered by Vimala Ramamurthy | 27th May, 2014, 09:08: AM