If-f-x-and-g-x-have-no-constant-term-then-f-x-g-x-f-x-g-x-

Question Number 2005 by Rasheed Soomro last updated on 29/Oct/15

I f f (x) a n d g (x) h a v e n o c o n s t a n t t e r m t h e n

f^{'} (x) = g^{'} (x) \overset{?}{\Rightarrow} f (x) = g (x) ?

Commented by prakash jain last updated on 30/Oct/15

If f (x) \neq g (x)

since both f (x) and g (x) have no constant

term . f (x) - g (x) = h (x) \neq c

f^{'} (x) - g^{'} (x) = h^{'} (x) \neq 0

∴ f (x) = g (x)

Commented by Rasheed Soomro last updated on 31/Oct/15

V e r y N i c e!

Answered by Filup last updated on 29/Oct/15

No constant, so in form :

f (x) = {a x}^{n}

g (x) = {b x}^{m}

f^{'} (x) = {a n x}^{n - 1}

g^{'} (x) = {b m x}^{m - 1}

f (x) = g (x)

{a x}^{n} = {b x}^{m}

∴ {({a x}^{n})}^{'} = {({b x}^{m})}^{'}

{a n x}^{n - 1} = {b m x}^{m - 1}

∴ Iff a n = b m and n = m will your

statement be correct

(S o r r y i f t h i s i s i n c o r r e c t)

T h i s i s o n l y o n e s o l u t i o n . I n e g l e c t e d

t h e u s e o f f u n c t i o n s u c h a s s i n e c o s i n e a n d t a n g e n t

Commented by Rasheed Soomro last updated on 29/Oct/15

{THANK}^{S} t o a t t e m p t t h e q u e s t i o n . Y o u h a v e t a k e n

o n l y a s p e c i a l c a s e o f p o l y n o m i a l (M o n o m i a l) .

I n c a s e o f p o l y n o m i a l f (x) a n d g (x) m a y b e t a k e n i n

g e n e r a l f o r m :

f (x) = \underset{i = 1}{\overset{n}{Σ}} a_{i} x^{i} a n d g (x) = \underset{i = 1}{\overset{n}{Σ}} b_{i} x^{i} p r e v e n t e d i t o b e

z e r o i n o r d e r t o a v o i d c o n s t a n t t e r m .