A calculus problem by vishnu c

Calculus Level 5

Let $f(x) = (x-a)(x-b)^3 (x-c)^5 (x-d)^7$ , where $a,b,c,d$ real numbers with $a<b<c<d$ .

Consider the derivative of $f(x)$ , $f'(x)$ . Find the number of distinct real roots of $f'(x) = 0$ .