1.

2x^(2)

Answer»

`6x-4`
`X^(2)+3x-2`
`-x^(2)+3x-2`
`-x^(2)+9x-6`

Solution :We have
`f(x+y)=f(x)+2y^(2)+"kxy for all "x,y in R`
`Rightarrow (f(x+y)-f(x))/(y)=2y"+kx for all "x inR`
`Rightarrow UNDERSET(y to 0)lim (f(x+y)-f(x))/(y)=underset(y to 0)lim (2y+kx)`
`Rightarrow underset(y to 0)lim (f(x+y)-f(y))/(y)=underset(y to 0)lim (2y+kx)`
`Rightarrow f'(x)=kx" for all "x in R`
`Rightarrow f'(x)=kx" for all "x in R`
But, f(1)=2 and f(2)=8. Therefore,
`2=(k)/(2)+Cand 8=2k+C`
`Rightarrow k=4 and C=0`
HENCE, f(x)=`2x^(2)"for all "x in R`


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