Let F(x) = X^2 - 2x And G(x) = F(f(x) - 1) + F(5 - F(x)), Then - Toppr

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SolveGuidesJoin / LoginUse appLogin0You visited us 0 times! Enjoying our articles? Unlock Full Access!Standard XIIMathematicsQuestionLet f(x)=x2−2x and g(x)=f(f(x)−1)+f(5−f(x)), then
  1. g(x)<0,∀x∈R
  2. g(x)<0 for some x∈R
  3. g(x)≤0 for some x∈R
  4. g(x)≥0,∀x∈R
Ag(x)<0,∀x∈RBg(x)≤0 for some x∈RCg(x)<0 for some x∈RDg(x)≥0,∀x∈ROpen in AppSolutionVerified by Toppr

Given, f(x)=x2−2xNow, f(f(x)−1)=f(x2−2x−1)=(x2−2x−1)2−2(x2−2x−1)And, f(5−f(x))=f(5−x2+2x)=(5−x2+2x)2−2(5−x2+2x)Hence, g(x)=f(f(x)−1)+f(5−f(x))∴g(x)=(x2−2x−1)2−2(x2−2x−1)+(5−x2+2x)2−2(5−x2+2x)⇒g(x)=(x2−2x−1)2+(5−x2+2x)2−2(x2−2x−1+5−x2+2x)∴g(x)=(x2+2x−1)2+(5−2x2+2x2)2−8Since the first two terms are in square,∴ it can not be negative and if x = 9 then we also get positive value.∴g(x)≥0∀x∈R∴ option D is correct

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