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2x^(2) |
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Answer» `6x-4` `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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