The curvature of a graph at a point for a smooth function \( f : (a,b) \to \mathbb{R} \) is defined as \( \frac{f''(x_0)}{(1 + (f'(x_0))^2)^{3/2}} \) for any \( x_0 \) within the interval \( (a,b) \). This formula derives from the second derivative of the function, \( f''(x_0) \), and incorporates the first derivative, \( f'(x_0) \), to account for the slope of the tangent line. Understanding this curvature formula is essential for analyzing the geometric properties of the graph of the function.
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