JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Suppose a differentiable function \(f ( x )\) satisfies the identity \(f(x+y)=f(x)+f(y)+x y^{2}+x^{2} y\) for all real \(x\) and \(y .\) If \(\lim \limits_{x \rightarrow 0} \frac{f(x)}{x}=1,\) then \(f^{\prime}(3)\) is equal to
- A \(8\)
- B \(9\)
- C \(10\)
- D \(12\)
Answer & Solution
Correct Answer
(C) \(10\)
Step-by-step Solution
Detailed explanation
since, \(\lim _{x \rightarrow 0} \frac{f(x)}{x}\) exist \(\Rightarrow f(0)=0\) Now, \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) \(=\lim _{h \rightarrow 0} \frac{f(h)+x h^{2}+x^{2} h}{h}(\operatorname{tak}(y=h)\)…
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