यदि `f((x+y)/2) =(f(x)+f(y))/2 AA x, y`
यदि `f'(0)=-1` तथा `f(0)=1` तब `f(3)` का मान ज्ञात कीजिए।

Explanation

दिया है: `f((x+y)/2)=(f(x)+f(y))/2`
`implies f((x+0)/2)=(f(x)+f(0))/2`
`implies f(x/2)=1/2 [f(x)+f(0)]=1/2 [f(x)+1]`
`implies f(x)=2f(x/2)-1` ...(1)
अब `f'(x)=lim_(h rarr 0) (f(x+h)-f(x))/h`
`=lim_(h rarr 0) {(f((2x+2h)/2)-f(x))/h}`
`=lim_(h rarr 0) (1/2 [f(2x)+f(2h)]-f(x))/h`
`=lim_(h rarr 0) (f(x+h)-f(x))/h` [समीकरण (1) से]
`=lim_(h rarr 0) (f(0+h)-f(0))/h=f'(0)=-1`
`implies f'(x)=-1`
`implies f(x)=-x+c` ...(2)
`x=0` रखने पर,
`f(0)=-0+c`
`implies c=f(0)=1` (दिया है)
`implies c=1`
`:. f(3)=-3+1=-2`