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

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.