exercises_12 - , 3] R given by f ( x ) = x ( x-2)( x-4) ....

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Analysis 1: Exercises 12 1. Let f be a continuous function on [ a, b ]. Denote M := max [ a,b ] f, m := min [ a,b ] f. Prove that ( c ( m, M )) ( x 0 ( a, b ) f ( x 0 ) = c ² . 2. Let f and g be differentiable at a . Prove that f + g is also differentiable at a , and ( f + g ) 0 ( a ) = f 0 ( a ) + g 0 ( a ) . 3. Let n N . Let f : R R given by f ( x ) = x n . Prove by induction that for all n N the function f is differentiable at any a R and f 0 ( a ) = na n - 1 . 4. Prove that if f : R R is an even (resp. odd) function and has a derivative at every point, then the derivative f 0 is an odd (resp. even) function. 5. Prove that, if a polynomial P ( x ) is divisible by ( x - a ) 2 , then P 0 ( x ) is divisible by ( x - a ). 6. (a) Let f ( x ) = xg ( x ) for some function g which is continuous at 0. Prove that f is differentiable at 0, and find f 0 (0) in terms of g . (b) Prove that the function f : R R given by f ( x ) = x | x | is differentiable for all x R and compute its derivative. 7. (a) Verify the validity of Rolle’s theorem for the function f : [1 , 3] R given by f ( x ) = ( x - 1)( x - 2)( x - 3) . (b) Verify the validity of the Mean Value Theorem for the function f : [1
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Unformatted text preview: , 3] R given by f ( x ) = x ( x-2)( x-4) . 8. Let a 1 , a 2 , ..., a n be real numbers and let f be dened on R by f ( x ) := n X i =1 ( a i-x ) 2 for x R . Find the unique point of relative minimum for f . 9. Let f, g be dierentiable on R and suppose that f (0) = g (0) and that f ( x ) 6 g ( x ) for all x > 0. Show that f ( x ) 6 g ( x ) for all x > 0. 10. Prove that the polynomial f m ( x ) = x 3-3 x + m never has two roots in [0 , 1], no matter what m is. 2 11. Let f be continuous on [ a,b ] and dierentiable on ( a,b ). Suppose that f 2 ( b )-f 2 ( a ) = b 2-a 2 . Prove (using Rolles theorem) that ( x ( a,b ) )( f ( x ) f ( x ) = x ) . 12. Using the Mean Value Theorem prove that ( (-1 , 0) ) 1 + 2 1 + < 1 + < 1 + 2...
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exercises_12 - , 3] R given by f ( x ) = x ( x-2)( x-4) ....

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