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Let f(x)=(x−a)(x−b)3(x−c)5(x−d)7f(x) = (x-a)(x-b)^3 (x-c)^5 (x-d)^7 f(x)=(x−a)(x−b)3(x−c)5(x−d)7, where a,b,c,da,b,c,da,b,c,d real numbers with a<b<c<da<b<c<da<b<c<d.
Consider the derivative of f(x)f(x)f(x), f′(x)f'(x)f′(x). Find the number of distinct real roots of f′(x)=0f'(x) = 0 f′(x)=0.
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