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How to solve this question wihout drawing graph?

Asked by Vineeth K 31st May 2016, 3:35 PM
Answered by Expert
Answer:
C o n s i d e r space t h e space g i v e n space p o l y n o m i a l comma
f open parentheses x close parentheses equals x to the power of 4 plus x cubed minus 5 x squared plus 2 space a n d space g open parentheses x close parentheses equals 2 x plus 3

C a s e space f o r space t h e space o p t i o n space B colon
G i v e n space t h a t space f open parentheses x close parentheses equals g open parentheses x close parentheses space f o r space e x a c t l y space 3 space v a l u e s space o f space x
G i v e n space p o l y n o m i a l space i s space o f space d e g r e e space 4 space a n d space h e n c e space t h e r e space a r e space 4 space r o o t s space o f space t h e space p o l y n o m i a l.
W e space k n o w space t h a t comma space i m a g i n a r y space r o o t s space o c c u r space i n space p a i r s space a n d space h e n c e space w e space s h o u l d space h a v e space space 2 space r e a l space r o o t s
o r space 4 space r e a l space r o o t s space o r space n o space r e a l space r o o t s.
T h u s comma space t h e space e q u a t i o n space f open parentheses x close parentheses equals g open parentheses x close parentheses space c a n n o t space h a v e space 3 space r e a l space r o o t s
H e n c e space o p t i o n space B space i s space i n c o r r e c t

C a s e space f o r space t h e space o p t i o n space C colon
G i v e n space t h a t space g open parentheses x close parentheses equals 0 space f o r space e x a c t l y space 4 space v a l u e s space o f space x
G i v e n space t h a t space g open parentheses x close parentheses space i s space a space l i n e a r space f u n c t i o n space o f space x space a n d space h e n c e space i t space s h o u l d space h a v e space o n l y space o n e space r e a l space r o o t
H e n c e space t h e space e q u a t i o n space space g open parentheses x close parentheses equals 0 space c a n n o t space h a v e space 4 space v a l u e s.
H e n c e space o p t i o n space C space i s space i n c o r r e c t

C a s e space f o r space t h e space o p t i o n space D colon
G i v e n space t h a t space f open parentheses x close parentheses equals 0 space f o r space e x a c t l y space 3 space v a l u e s space o f space x
G i v e n space p o l y n o m i a l space i s space o f space d e g r e e space 4 space a n d space h e n c e space t h e r e space a r e space 4 space r o o t s space o f space t h e space p o l y n o m i a l.
W e space k n o w space t h a t comma space i m a g i n a r y space r o o t s space o c c u r space i n space p a i r s space a n d space h e n c e space w e space s h o u l d space h a v e space space 2 space r e a l space r o o t s
o r space 4 space r e a l space r o o t s space o r space n o space r e a l space r o o t s.
T h u s comma space t h e space e q u a t i o n space f open parentheses x close parentheses equals 0 space c a n n o t space h a v e space 3 space r e a l space r o o t s
H e n c e space o p t i o n space D space i s space i n c o r r e c t

C a s e space f o r space t h e space o p t i o n space E
C o n s i d e r space t h e space e q u a t i o n space f open parentheses x close parentheses equals g open parentheses x close parentheses
rightwards double arrow x to the power of 4 plus x cubed minus 5 x squared plus 2 space equals 2 x plus 3
rightwards double arrow x to the power of 4 plus x cubed minus 5 x squared plus 2 space minus 2 x minus 3 equals 0
rightwards double arrow x to the power of 4 plus x cubed minus 5 x squared space minus 2 x minus 1 equals 0
L e t space h open parentheses x close parentheses equals f open parentheses x close parentheses minus g open parentheses x close parentheses
C o n s i d e r space h open parentheses negative 2 close parentheses equals open parentheses negative 2 close parentheses to the power of 4 plus open parentheses negative 2 close parentheses cubed minus 5 open parentheses negative 2 close parentheses squared space minus 2 open parentheses negative 2 close parentheses minus 1
rightwards double arrow h open parentheses negative 2 close parentheses equals 16 minus 8 minus 20 plus 4 minus 1 equals negative 9 equals negative v e

A n d space C o n s i d e r space h open parentheses 2 close parentheses equals open parentheses 2 close parentheses to the power of 4 plus open parentheses 2 close parentheses cubed minus 5 open parentheses 2 close parentheses squared space minus 2 open parentheses 2 close parentheses minus 1
rightwards double arrow h open parentheses negative 2 close parentheses equals 16 plus 8 minus 20 minus 4 minus 1 equals negative 1 equals negative v e
T h u s comma space t h e r e space i s space n o space c h a n g e space o f space s i g n space i n space h open parentheses negative 2 close parentheses space a n d space h open parentheses 2 close parentheses
rightwards double arrow T h e r e space i s space n o space r o o t space b e t w e e n space minus 2 space a n d space 2 space s u c h space t h a t space f open parentheses x close parentheses equals g open parentheses x close parentheses
rightwards double arrow T h e r e f o r e comma space f open parentheses x close parentheses not equal to g open parentheses x close parentheses space f o r space a n y space v a l u e space o f space x space x space w h e n space minus 2 less than x less than 2
H e n c e space o p t i o n space E space i s space c o r r e c t.

C a s e space f o r space o p t i o n space A colon
C o n s i d e r space t h e space e q u a t i o n space f open parentheses x close parentheses equals g open parentheses x close parentheses
rightwards double arrow x to the power of 4 plus x cubed minus 5 x squared plus 2 space equals 2 x plus 3
rightwards double arrow x to the power of 4 plus x cubed minus 5 x squared plus 2 space minus 2 x minus 3 equals 0
rightwards double arrow x to the power of 4 plus x cubed minus 5 x squared space minus 2 x minus 1 equals 0
L e t space h open parentheses x close parentheses equals f open parentheses x close parentheses minus g open parentheses x close parentheses
C o n s i d e r space h open parentheses negative 2 close parentheses equals open parentheses negative 2 close parentheses to the power of 4 plus open parentheses negative 2 close parentheses cubed minus 5 open parentheses negative 2 close parentheses squared space minus 2 open parentheses negative 2 close parentheses minus 1
rightwards double arrow h open parentheses negative 2 close parentheses equals 16 minus 8 minus 20 plus 4 minus 1 equals negative 9 equals negative v e

A n d space C o n s i d e r space h open parentheses negative 3 close parentheses equals open parentheses negative 3 close parentheses to the power of 4 plus open parentheses negative 3 close parentheses cubed minus 5 open parentheses negative 3 close parentheses squared space minus 2 open parentheses negative 3 close parentheses minus 1
rightwards double arrow h open parentheses negative 3 close parentheses equals 81 minus 27 minus 45 plus 6 minus 1 equals 14 equals plus v e
T h u s comma space t h e r e space i s space a space c h a n g e space o f space s i g n space i n space h open parentheses negative 2 close parentheses space a n d space h open parentheses negative 3 close parentheses
rightwards double arrow T h e r e space i s space a space r o o t space b e t w e e n space minus 2 space a n d space minus 3 space s u c h space t h a t space f open parentheses x close parentheses equals g open parentheses x close parentheses

C o n s i d e r space h open parentheses 2 close parentheses equals open parentheses 2 close parentheses to the power of 4 plus open parentheses 2 close parentheses cubed minus 5 open parentheses 2 close parentheses squared space minus 2 open parentheses 2 close parentheses minus 1
rightwards double arrow h open parentheses 2 close parentheses equals 16 plus 8 minus 20 minus 4 minus 1 equals negative 1 equals negative v e

A n d space C o n s i d e r space h open parentheses 3 close parentheses equals open parentheses 3 close parentheses to the power of 4 plus open parentheses 3 close parentheses cubed minus 5 open parentheses 3 close parentheses squared space minus 2 open parentheses 3 close parentheses minus 1
rightwards double arrow h open parentheses 3 close parentheses equals 81 plus 27 minus 45 minus 6 minus 1 equals 56 equals plus v e
T h u s comma space t h e r e space i s space a space c h a n g e space o f space s i g n space i n space h open parentheses 2 close parentheses space a n d space h open parentheses 3 close parentheses
rightwards double arrow T h e r e space i s space a space r o o t space b e t w e e n space 2 space a n d space 3 space s u c h space t h a t space f open parentheses x close parentheses equals g open parentheses x close parentheses
rightwards double arrow T h e r e space a r e space t w o space r e a l space r o o t s space s u c h space t h a t space f open parentheses x close parentheses equals g open parentheses x close parentheses

W e space k n o w space t h a t comma space i m a g i n a r y space r o o t s space o c c u r space i n space p a i r s space a n d space h e n c e space w e space s h o u l d space h a v e space space 2 space r e a l space r o o t s
o r space 4 space r e a l space r o o t s.
A n d space C o n s i d e r space h open parentheses 4 close parentheses equals open parentheses 4 close parentheses to the power of 4 plus open parentheses 4 close parentheses cubed minus 5 open parentheses 4 close parentheses squared space minus 2 open parentheses 4 close parentheses minus 1
rightwards double arrow h open parentheses 4 close parentheses equals 256 plus 64 minus 80 minus 8 minus 1 equals plus v e
T h u s comma space t h e r e space i s space n o space c h a n g e space o f space s i g n space i n space h open parentheses 3 close parentheses space a n d space h open parentheses 4 close parentheses
rightwards double arrow A n d space i f space x greater than 4 comma space h left parenthesis x right parenthesis space i s space a l w a y s space g r e a t e r space t h a n space 0 space

A n d space C o n s i d e r space h open parentheses negative 4 close parentheses equals open parentheses 4 close parentheses to the power of 4 plus open parentheses negative 4 close parentheses cubed minus 5 open parentheses negative 4 close parentheses squared space minus 2 open parentheses negative 4 close parentheses minus 1
rightwards double arrow h open parentheses negative 4 close parentheses equals 256 minus 64 minus 80 plus 8 minus 1 equals plus v e
T h u s comma space t h e r e space i s space n o space c h a n g e space o f space s i g n space i n space h open parentheses negative 3 close parentheses space a n d space h open parentheses negative 4 close parentheses
rightwards double arrow A n d space i f space x less than negative 3 comma space h left parenthesis x right parenthesis space i s space a l w a y s space g r e a t e r space t h a n space 0 space

T h e r e f o r e t h e r e space a r e space n o space c h a n g e s space o f space s i g n space i f space x less than negative 3 space a n d space i f space x greater than 4
T h u s space t h e r e space a r e space o n l y space t w o space r e a l space r o o t s space f o r space w h i c h space f open parentheses x close parentheses equals g open parentheses x close parentheses
H e n c e space o p t i o n space A space i s space i n c o r r e c t.








Answered by Expert 1st June 2016, 5:11 PM
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