(Question 7 continued)
In the following diagram, the line $L$ is normal to the graph of $f$ at point $\mathrm{P}$.
(d) Find the equation of the line $L$, in the form $y=a x+b$.
The line $L$ intersects the graph of $f$ at a second point, $Q$, as shown above.
(e) Calculate the distance between $\mathrm{P}$ and $\mathrm{Q}$.
Answer
The second point Q is the same as point P, so the distance between P and Q is \(\boxed{0}\)
Step 1 ：Find the derivative of the function f(x) and evaluate it at x = 2: \(f'(x) = 3x^2 - 12x + 12\), \(f'(2) = 0\)
Step 2 ：Since the slope of the tangent line at point P is 0, the equation of the line L is \(y = 0\)
Step 3 ：Solve the equation f(x) = 0: \(x^3 - 6x^2 + 12x - 8 = 0\), x = 2
Step 4 ：The second point Q is the same as point P, so the distance between P and Q is \(\boxed{0}\)
