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We know that for some well-behaved, smooth/continuous, twice differentiable function of x, f[x] there exists at each point a slope (f ' [x]) and a radius of curvature

[tex]\rho [x]=\frac{\left(1+f'[x]^2\right)^{\frac{3}{2}}}{f\text{''}[x]}[/tex]

It also seems intuitive to think that at every point on such a function, the tangent line vector and the vector from the center of curvature to this point would be normal to one another.

Is this necessarily true?