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Let f\left(\frac{x+y}{2}\right)=\frac{1}{2}(f(x)+f(y)) for real x and y. If f'(0) exists and equals to -1and f(0)=1 then the value of f(2) is

Option: 1

1


Option: 2

-1


Option: 3

1/2


Option: 4

2


Answers (1)

best_answer

\begin{aligned} \mathrm{f}^{\prime}(x) & =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \\ & =\lim _{h \rightarrow 0} \frac{\frac{f(2 x)+f(2 h)}{2}-f(x)}{h} \\ \mathrm{f}^{\prime}(x) & =-1 \quad ; f(2 x)=2 f(x)-1 \\ \Rightarrow f(x) & =1-x \quad \end{aligned}

 

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SANGALDEEP SINGH

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