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JEE Main 2021
Differentiation
Differentiation
Medium

Question

Let f:RRf:\mathbb{R}\to\mathbb{R} be a differentiable function that satisfies the relation f(x+y)=f(x)+f(y)1,x,yRf(x+y)=f(x)+f(y)-1,\forall x,y\in\mathbb{R}. If f(0)=2f'(0)=2, then f(2)|f(-2)| is equal to ___________.

Answer: 1

Solution

f(x+y)=f(x)+f(y)1f(x+y)=f(x)+f(y)-1 f(x)=limh0f(x+h)f(x)hf(x)=limh0f(h)f(0)h=f(0)=2f(x)=2dy=2dxy=2x+Cx=0,y=1,c=1y=2x+1f(2)=4+1=3=3\begin{aligned} & f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\\\ & f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(h)-f(0)}{h}=f^{\prime}(0)=2 \\\\ & f^{\prime}(x)=2 \Rightarrow d y=2 d x \\\\ & y=2 x+C \\\\ & \mathrm{x}=0, \mathrm{y}=1, \mathrm{c}=1 \\\\ & \mathrm{y}=2 \mathrm{x}+1 \\\\ & |f(-2)|=|-4+1|=|-3|=3 \end{aligned}

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