Question:
Suppose a differentiable function \(f(x)\) satisfies the identity \(f(x+y)=f(x)+f(y)+xy^2+x^2y\), for all real \(x\) and \(y\). If \(\lim_{x\to 0}\frac{f(x)}{x}=1\), then \(f'(3)\) is equal to ______.
Suppose a differentiable function \(f(x)\) satisfies the identity \(f(x+y)=f(x)+f(y)+xy^2+x^2y\), for all real \(x\) and \(y\). If \(\lim_{x\to 0}\frac{f(x)}{x}=1\), then \(f'(3)\) is equal to ______.
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Use the definition of the derivative with the given identity; \(f(0)=0\) follows by putting \(x=y=0\).
Updated On: Nov 5, 2026
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Correct Answer: 10
Solution and Explanation
Putting \(x=y=0\): \(f(0)=2f(0)\Rightarrow f(0)=0\).
\(f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{f(h)+xh^2+x^2h}{h}=\lim_{h\to 0}\left[\frac{f(h)}{h}+xh+x^2\right]=1+x^2\).
\(f'(3)=1+9=10\).
Answer: 10
\(f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{h\to 0}\frac{f(h)+xh^2+x^2h}{h}=\lim_{h\to 0}\left[\frac{f(h)}{h}+xh+x^2\right]=1+x^2\).
\(f'(3)=1+9=10\).
Answer: 10
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