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  • Suppose differntial function f(x) satisfies the identity ,

Suppose differntial function f(x) satisfies the identity f(x+y)=f(x)+f(y)+xy^{2}+x^{2}y, for all real x any y. If \lim_{x\rightarrow 0}\frac{(x)}{x}=1 then f'(3) is equal to ________  
Option: 1 8
Option: 2 10
Option: 3 12
Option: 4 14

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best_answer

\\\text { Since, } \lim _{x \rightarrow 0} \frac{f(x)}{x} \text { exist } \Rightarrow f(0)=0 \\ \text { Now, } f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ =\lim _{h \rightarrow 0} \frac{f(h)+x h^{2}+x^{2} h}{h}(\text { take } y=h)

\\=\lim _{h \rightarrow 0} \frac{f(h)}{h}+\lim _{h \rightarrow 0}(x h)+x^{2} \\ \Rightarrow f^{\prime}(x)=1+0+x^{2} \\ \Rightarrow f^{\prime}(3)=10

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himanshu.meshram

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