Let's say f satisfies:

f''(x) + f'(x)g(x) - f(x) = 0

Show that iffis 0 at 2 points, thenfis 0 in some interval in between them

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How would you start approaching this problem? I tried just trying to do some manipulations with f(x) = f''(x) + f'(x)g(x) and in this case then if f(a) = f''(a) + f'(a)g(a) = 0 and f(b) = f''(b) + f'(b)g(b) = 0 and trying to find some intervals, but I haven't had much luck.

Any help is appreciated, thanks!