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Proving part b
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I assume you repeatedly apply Rolle's theorem, as for part a and count the number of zeroes, but I don't see why it requires N differentiations. Once you differentiate max{k_i} times, none of the a_i will be zeroes anymore.
(This is part of exercise 4.6.6 in "Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach" by Hubbard & Hubbard.)
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