Does this function have a unique tangent line at x = 0?
No. Imagine a line that intersects the graph only at the point x = 0 (for example, a horizontal line). If you push down on the line gently to the left or right of x = 0, the line will continue to touch only at the one point but will move to a different angle of inclination. Thus, there is no unique tangent line. This will be the situation at any corner point on the graph of a function.
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