Let for

Finding the Critical Point

By differentiating the function and setting , the critical point is found at

The Role of

If we evaluate the second derivative at , the result depends explicitly on the value and sign of the exponent m.

For the function to safely guarantee a true local minimum rather than a saddle point or maximum across all variations of parameters, the entire behavior of the function's boundary conditions, rates of change, and global extrema dynamically updates based on whether is a positive, negative, fractional, or even/odd value.

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