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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` |
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Answer» 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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