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Let say here the intersection point is 5:

f(g) is for range [0<n<=5]

and

f(x) is for range [5<=n<10]

for f(g) real root using quadratic equation is 4.3 that lies within its range and results in equation =0 however, the minimum value of the first derivative I got is n=5 instead of n=4.3. And it is always the case and vice versa for f(x). How do I prove that intersection point in the range is always be the minimum solution?