Answer to Question #188882 in Real Analysis for TUHIN SUBHRA DAS

Question #188882

Let f :[0,1] tends to R be a function defined by f(x)=x^m (1-x)^n, where m,n belongs to N.Find the values of m and n such that the Rolle’s Theorem holds for the function f .


1
Expert's answer
2021-05-07T13:09:59-0400

"\\forall n,m \\in N,f(x)"- Polynomial"\\implies" "f(x)" -  continuous and differentiable on "(-\\infty,\\infty)" "\\implies f(x)" is continuous on the closed interval [0, 1] and differentiable on the open interval (0, 1), "f(0)=0^m *1^n=0,f(1)=1^m*0^n=0,f(0)=f(1)" "\\implies" the Rolle’s Theorem holds for the function "f(x)" Answer:"\\forall n,m \\in N"

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