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Let f(x)=x^2-2x ,x in R ,a n dg(x)=f(f(...

Let `f(x)=x^2-2x ,x in R ,a n dg(x)=f(f(x)-1)+f(5-(x))dot` Show that `g(w)geqoAAx in Rdot`

Text Solution

Verified by Experts

Given `f(x)=x^(2)-2x =(x-1)^(2)-1`
or `g(x)=f(f(x)-1)+f(5-f(x))`
`=f((x-1)^(2)-2)+f(6-(x-1)^(2))`
`=[(x-1)^(2)-2-1]^(2)-1+[6-(x-1)^(2)-1]^(2)-1`
`=(x-1)^(4)-6(x-1)^(2)+9-1+(x-1)^(4)-10(x-1)^(2)+25-1`
`=2(x-1)^(4)-16(x-1)^(2)+32`
`=2[(x-1)^(4)-8(x-1)^(2)+16]`
`=2[(x-1)^(2)-4]^(2) ge 0 AA x in R`
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