A function I has derivatives of all orders for all real numbers , X . Assume f ( 2 ) = - 3 , f ( 2 ) = 5 , f ( 2 ) = 3 and the third...

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A function I has derivatives of all orders for all real numbers , X . Assume f ( 2 ) = - 3 , f " ( 2 ) = 5 , f " ( 2 ) = 3 and the third derivative of fat X = 2 is - 8 .1 . Write the 3 rd degree Taylor polynomial for & about * = 2 and use it to approximate $ ( 1. 5 ) .2 . The fourth derivative of { satisfies the inequality*$( 4 ) * < 3 for all x in the interval [1.5 , 21 . Use the La Grange error bound on the approximation to $ ( 1 . 5 ) found inpart ( a ) to explain why f ( 1 . 5 ) cannot equal - 5 .3 . Write a fourth degree Taylor polynomial P ( X ) , for g ( X ) = * ( * 2 + 2 ) about * = 0. Use P ( X) to explain why g must have a relative minimum at * = 0 .4 . Given the function f ( X) = e-2x` , find the first four non- zero terms and the general term of the power series for * ( * ) about * = 0 .5 . Find the interval of convergence of the power series for* ( * ) about * = 0 . Show the analysis that leads to your conclusion .Print