JEE Main Test 43

Result

JEE Main Test 43

Score
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Rank
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TIME Taken - -

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Question 1
4 / -1

Suppose ' f ' is continuous function from R to R and f ( f ( a ) ) = a for some a ∈ R then the equation f ( x ) = x has

A
exactly one solution

B
at most one solution

C
no solution

D
at least three solutions

Solution

If f ( a ) = a, then obviously x = a is the solution

Let f ( a ) > a and g ( x ) = f ( x ) - x

then g ( a ) > 0 and

g ( f ( a ) ) = f ( f ( a ) ) - f ( a ) = a - f ( a ) < 0

Since g(x) is continuous, so at least for one

c ∈ ( a, f ( a ) ), g ( c ) = 0.

Similarly we can argue for f ( a ) < a.

The correct answer is: at least three solutions