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Application Of Derivatives

Question
CBSEENMA12035248

Show that the function f(x) = x2 is a decreasing function in (– ∞  0).

Solution
Let x1, x2 ∊ (– ∞ , 0) and let x1 < x2.
        Now,       x1 < x2
             rightwards double arrow space space space space straight x subscript 1. space straight x subscript 1 space greater than space space straight x subscript 1. space straight x subscript 2                                         open square brackets because space space space straight x subscript 1 less than 0 close square brackets
              rightwards double arrow space space straight x subscript 1 squared space greater than space straight x subscript 1. space straight x subscript 2                                                             ...(1)
Again   straight x subscript 1 space less than space straight x subscript 2
          rightwards double arrow space space straight x subscript 1. space straight x subscript 2 space space greater than space space space straight x subscript 2. end subscript space straight x subscript 2                                          open square brackets because space space straight x subscript 2 less than 0 close square brackets
          rightwards double arrow space space space space space straight x subscript 1. space straight x subscript 2 space greater than space straight x subscript 2 squared                                                            ...(2)
From (1) and (2), we get,   straight x subscript 1 squared space greater than space straight x subscript 2 squared space space space space space space or space space space space straight f left parenthesis straight x subscript 1 right parenthesis thin space greater than space straight f left parenthesis straight x subscript 2 right parenthesis
therefore space space space straight x subscript 1 space less than space straight x subscript 2 space space space space space space rightwards double arrow space space space space straight f left parenthesis straight x subscript 1 right parenthesis space greater than space straight f left parenthesis straight x subscript 2 right parenthesis
therefore space space space space straight f space is space an space decreasing space function space in space left parenthesis negative infinity comma space 0 right parenthesis.