InterviewSolution
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InterviewSolution
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Let `f : [0, 1] rarr [0, 1] `be a continuous function such that `f (f (x))=1 for all x in[0,1]`then:A. `f(x)=x` for at least one `x in (0,1)`B. f(x) will be differential in [0,1]C. f(x)+x=0 for at least one x such that `0 le xle 1`D. none of these |
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Answer» Correct Answer - A Since `f:[0,1] to [0,1]` is a continuous function. Therefore, Range `(f) subset [0,1] and f(x)` attains every value between f(0) and f(1). Let g(x)=f(x)-x for all `x in [0,1]` Clearly, g(x) is also continuous on [0,1] Also, `g(0)=f(0)-0=f(0) gt 0` and `g(1)=f(1)-1=0" "[therefore f(1) lt 1]` Thus, g(x) is continuous on [0,1] such that g(0) g(1)`lt 0 ` . Therefore there exists a point `c in (0,1)` such that `g(c)=0 Rightarrow f(c )=c` Hence, f(x)=x for at least one `x in (0,1)` |
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