Key Concepts and Formulas

• Slope of a Tangent: The derivative of a function $y = f(x)$, denoted as $\frac{dy}{dx}$ or $f'(x)$, represents the slope of the tangent line to the curve at a given point $(x, y)$.
• Angle Between Two Lines: If two lines have slopes $m_1$ and $m_2$, the angle $\theta$ between them can be found using the formula: $\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$
• Perpendicular Lines: Two lines are perpendicular if and only if the product of their slopes is -1, i.e., $m_1 m_2 = -1$. In this case, the angle between the lines is $\frac{\pi}{2}$.

Step-by-Step Solution

1. Verify the Points Lie on the Curve We are given the curve $y = x^2 - 5x + 6$ and the points $(2, 0)$ and $(3, 0)$. We need to verify that these points actually lie on the curve.

• For $(2, 0)$: Substituting $x = 2$ into the equation gives $y = (2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0$. So, the point $(2, 0)$ lies on the curve.
• For $(3, 0)$: Substituting $x = 3$ into the equation gives $y = (3)^2 - 5(3) + 6 = 9 - 15 + 6 = 0$. So, the point $(3, 0)$ lies on the curve.

2. Find the Derivative of the Curve To find the slope of the tangent at any point on the curve, we need to find the derivative $\frac{dy}{dx}$. Given $y = x^2 - 5x + 6$, we differentiate with respect to $x$: $\frac{dy}{dx} = \frac{d}{dx}(x^2 - 5x + 6) = 2x - 5$

3. Calculate the Slope at the Point (2, 0) We substitute $x = 2$ into the derivative to find the slope $m_1$ of the tangent at $(2, 0)$: $m_1 = \left. \frac{dy}{dx} \right|_{x=2} = 2(2) - 5 = 4 - 5 = -1$

4. Calculate the Slope at the Point (3, 0) We substitute $x = 3$ into the derivative to find the slope $m_2$ of the tangent at $(3, 0)$: $m_2 = \left. \frac{dy}{dx} \right|_{x=3} = 2(3) - 5 = 6 - 5 = 1$

5. Determine the Angle Between the Tangents We have $m_1 = -1$ and $m_2 = 1$. The product of the slopes is $m_1 m_2 = (-1)(1) = -1$. Since the product of the slopes is -1, the tangents are perpendicular, and the angle between them is $\frac{\pi}{2}$.

Alternatively, using the formula for the angle between two lines: $\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| = \left| \frac{-1 - 1}{1 + (-1)(1)} \right| = \left| \frac{-2}{0} \right|$ Since the denominator is zero, $\tan \theta$ is undefined, which means $\theta = \frac{\pi}{2}$.

Common Mistakes & Tips

• Always verify that the given points lie on the curve before proceeding.
• Double-check the derivative calculation to avoid errors.
• Recognize that $m_1 m_2 = -1$ implies perpendicular lines and an angle of $\frac{\pi}{2}$.

Summary

We found the slopes of the tangents at the given points by first finding the derivative of the curve's equation. Then, we calculated the slopes $m_1 = -1$ and $m_2 = 1$. Since the product of these slopes is -1, the tangents are perpendicular, and the angle between them is $\frac{\pi}{2}$.

The final answer is $\boxed{{\pi \over 2}}$, which corresponds to option (B).