Show that 13+23+...+n3â=(1+2+...+n)â2 âfor every positive interger n
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Let b > 0 and f : [0, b] â R be continuous and f(b) = 0. Assume that f is differentiable on (0, b). Show: For every n â N, there exists some c â (0, b) such that cf â²(c) = ânf(c).
Hint: Apply Rolleâs theorem to the function g(x) = xn f(x).
Besure to show step by step work.