MAD4401 Quiz11 - x not equal to z that f x − z< x − z...

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MAD 4401 Quiz 11 1. Let f ( x ) be continuous and suppose that the sequence x 0 , f ( x 0 ), f ( f ( x 0 )), f ( f ( f ( x 0 ))), converges to z . Show that f ( z ) = z . This sequence can also be defined as x n { } n = 0 where x 0 is arbitrary and x n + 1 = f ( x n ) for n = 0,1,2,3, . 2. State the Mean Value Theorem. 3. Let f : R R be a differentiable function. Suppose that f ( z ) = z is a fixed point for f . Suppose that for some ε > 0 for all c [ z , z + ] it is true that f ( c ) < 1 . Show that for any
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Unformatted text preview: x not equal to z that f ( x ) − z < x − z . 4. Consider the equation f ( x ) = and suppose that z is a solution. Show that z is a fixed point of the function g ( x ) = x − f ( x ) ′ f ( x ) . 5. Compute the derivative of g ( x ) = x − f ( x ) ′ f ( x ) . What is the value of ′ g ( z ) ?...
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