sheet6 - f x = cf x for all x ∈ R then f x = e cx f(0 for...

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CM221A ANALYSIS I Exercise Sheet 6 1. Find the maximum and minimum values of the function f ( x ) = 2 x 5 - x 3 on the interval [0 , 1], and sketch the graph of the function on this interval. 2. Prove that the derivative of the function f ( x ) = (sin x ) 2 + x 2 only vanishes at one point in [ - π, π ]. Find the maximum and minimum values of the function f ( x ) on the interval and sketch its graph. 3. Suppose that f is differentiable on the interval ( a, b ). Prove that if f 0 ( x ) > 0 for all x ( a, b ) then f is strictly monotonically increasing in the sense that f ( u ) > f ( v ) whenever u > v . Write down an example that proves the converse statement is false: there exists a differentiable function f such that f ( u ) > f ( v ) whenever u > v but f 0 ( x ) > 0 for all x is false. 4. By applying the mean value theorem to f ( x ) e - cx , prove that if f is differ- entiable on
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Unformatted text preview: f ( x ) = cf ( x ) for all x ∈ R then f ( x ) = e cx f (0) for all x ∈ R . 5. Using the identities (sin x ) = cos x and (cos x ) =-sin x , find the derivative of tan x . Use the chain rule and the identity arctan(tan x ) = x to show that the derivative of f ( x ) = arctan x is equal to (1 + x 2 )-1 . 6. Use the mean value theorem to obtain the best upper and lower bounds on arctan(1 . 1) that you can. Note that arctan(1) = π/ 4. 7. Use an appropriate theorem to find the range of values of x ∈ R for which each of the following power series converges: ( i ) ∞ X n =1 n 9-1 n 8 + 1 x n . ( ii ) ∞ X n =0 2 n + 3 n 4 n + 5 n x n . ( iii ) ∞ X n =0 (2 n )! x n ( n !) 2 . 1...
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