$$ \mathrm{Q}[3] $$ : Let $$ L $$ be the tangent line to a curve $$ y=f(x) $$ at some point $$ P $$. How many times will $$ L $$ intersect the curve $$ y=f(x) $$ ?

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1. The tangent line $$ L $$ to the curve $$ y=f(x) $$ at a point $$ P=(a,f(a)) $$ is given by
   $$
   L: \quad y = f(a) + f'(a)(x-a).
   $$
2. By definition, $$ L $$ touches the curve at $$ P $$, so when we set
   $$
   f(x) = f(a) + f'(a)(x-a),
   $$
   the equation is satisfied at $$ x=a $$.

3. In the case where $$ f $$ is a linear function, say $$ f(x)=mx+b $$, the tangent line at any point is the function itself. Hence,
   $$
   f(x) = mx+b = f(a) + m(x-a),
   $$
   and $$ L $$ and the curve coincide. Thus, they intersect at infinitely many points.

4. For a nonlinear function $$ f(x) $$, the equation
   $$
   f(x) - f(a) - f'(a)(x-a) = 0
   $$
   always has the solution $$ x=a $$ (which is the point of tangency). Whether it has other solutions depends on the form of $$ f $$.  
   - Often, for a smooth nonlinear function, the tangent line touches the curve only at $$ P $$ and does not meet it anywhere else.
   - However, there are cases where this equation has another solution, meaning $$ L $$ will cross the curve at another point apart from $$ P $$.

5. In summary, the answer depends on the nature of $$ f $$:
   - If $$ f $$ is linear, $$ L $$ and $$ y=f(x) $$ coincide and intersect in infinitely many points.
   - If $$ f $$ is nonlinear, $$ L $$ always intersects at least at $$ P $$, and may intersect the curve at a second point depending on the specific function.

Thus, the number of intersections is not fixed by the fact that $$ L $$ is tangent to $$ y=f(x) $$; it can be one, two, or infinitely many, depending on whether $$ f $$ is nonlinear or linear, respectively.
The tangent line $$ L $$ to the curve $$ y=f(x) $$ at point $$ P $$ intersects the curve at $$ P $$ and may intersect it at another point depending on the function $$ f $$. If $$ f $$ is linear, $$ L $$ coincides with the curve, resulting in infinitely many intersections. If $$ f $$ is nonlinear, $$ L $$ intersects the curve at least at $$ P $$ and possibly at one or more additional points.

\( \mathrm{Q}[3] \) : Let \( L \) be the tangent line to a curve \( y=f(x) \) at some point \( P \). How many times will \( L \) intersect the curve \( y=f(x) \) ?

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