IIT JEE , AIEEE Forum - Mathematics , Algebra

vineetvsb

1.if f(x)=(x²+1+x^-2)/(1+x^-1+x^-2),then d2y/dx2(2^1/2)=

2.if g(x)=(x^2+2x+3) f(x),f(0)=5 & lim _x_à0 [f(x)-5]/x=4 then g'(0)=

3.if f(x) is a polynomial satisfying f(x^2+1)=[f(x)]^2+1 &f(0)=0 ,then find f '(0)=

plz explain these questions?

vineetvsb

is there no one yo solve it?????????

vasanth

for question 3)

wat is that question mark for

vineetvsb

sorry, ok now i hav written the correct question

vinu

Hi vineet,

4 da 1st Q ;

Write,(x⁴+x²+1) = (x⁴+2x²+1) - (x²) ;

= (x²+1)² - (x)² ;

= (x²+1-x)(x²+1+x) ;

So, f(x) = (x⁴+x²+1)/(x²+x+1) = (x²+1-x) , now u can double differenciate it..... & put x =

2.

vinu

4 da 2nd Q ;

u've _[x]

_[0] {f(x) - 5} / x as 0/0 form ;

Apply l'hospital's rule....;

Then, f ' (x) = 4 .

g(x) = (x²+2x+3)*f(x) ;

g ' (x) = (x²+2x+3)* f ' (x) + f(x)*(2x+2) ; {put x= 0} ;

g ' (0) = 3(4) + 5(2) = 22 .{given f(0) = 5}

Moderator

Experts only answer one question at a time, please let us know which question needs to be answered first and post other queries again.

~ moderator

yahiyafirdous

Q1

f(x)=(x⁴+x²+1)/(x²+x+1) by multipkying both numerator and denomirator by x²

=(x²+x+1)(x²-x+1))/(x²+x+1)=x²-x+1

d²y/dx²=2

Q2

g^'(x)=_{[ h][ 0]} [ {(x+h)²+2(x+h)+3}f(x+h) - (x²+2x+3)f(x)]/h

g^'(0)=_{[ h][ 0]} [(h²+2h+3)f(h) - 3f(0)]/h

g^'(0)=_{[ h][ 0]} [(h²+2h+3)f(h) -15]/h Since f(0)=5

g^'(0)=_{[ h][ 0]} [(h²+2h)(f(h))/h+ (3f(h) -15)/h]

g^'(0)=_{[ h][ 0]} [(h+2)f(h) + 3(f(h) -5)/h]=2f(0)+ 3X4 = 10 + 12= 22

NOTE: _{[ h][ 0]} 3(f(h) - 5)/h =4

Q3

Since f(x²+1)=[f(x)]²+1 is true for all values of x, hence it also true for f(x)=x

which is obvious from the given equation.

f^'(x)=_{[ h][ 0]}[f(x+h) - f(x)]/h

f'(0)=_{[ h][ 0]}[f(h) - f(0)]/h

But f(0)=0 and f(h)=h

f'(0)=_{[ h][ 0]}h/h=1 Ans

iitkgp_bipin

f(x) = (x²+1+x^-2)/(1+x^-1+x^-2)

Multiplying numerator and denominator by x² we get

f(x) = (x⁴+x²+1)/(x²+x+1)
= (x²+x+1)(x²-x+1)/(x²+x+1)
= (x²-x+1)

Hence f"(x) = 2

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