Let . Using Rolle's theorem, prove that is a polynomial of degree n, with distinct roots, all of which lie in the interval (-1,1).
Can anyone help to get me started with this question?
If with satisfying the conditions of Rolle's Theorem we have now we observe that between two roots of there lies a root of . Now observe that has a root at in fact c=0. Now for a roots between (-1,0) and (0,1). Now is indeed an polynomial of order n now use induction to prove that it has distinct roots between (-1,1).
Kalyan.
For any is . The function is 'even' , i.e. is , and that means that is 'odd' , i.e. is , so that is and that is also consequence of the Rolle's theorem. The vanishes in , and and You can repeat the procedure for in to in and demonstrating that has a zero in each interval. Of course the same is for the derivatives of higher order...
Kind regards