Question:

Given the function $y = x^3 - 3x^2 + 6x + 2009$. **Step 1: Find the first derivative.** The first derivative of the function is: $$ \frac{dy}{dx} = 3x^2 - 6x + 6 $$ **Step 2: Find the second derivative.** The second derivative of the function is: $$ \frac{d^2y}{dx^2} = 6x - 6 $$ **Step 3: Determine the potential point of inflection.** A point of inflection occurs where the second derivative is equal to zero. Setting the second derivative to zero: $$ 6x - 6 = 0 $$ $$ 6x = 6 $$ $$ x = 1 $$ **Step 4: Verify the point of inflection.** To confirm that $x=1$ is a point of inflection, we examine the sign of the second derivative around $x=1$. For $x < 1$, $\frac{d^2y}{dx^2} = 6x - 6 < 0$, indicating the function is concave down. For $x > 1$, $\frac{d^2y}{dx^2} = 6x - 6 > 0$, indicating the function is concave up. Since the concavity changes at $x=1$, it is a point of inflection. **Step 5: Calculate the slope of the tangent at the point of inflection.** The slope of the tangent at the point of inflection is the value of the first derivative evaluated at $x=1$. $$ \text{Slope} = \left.\frac{dy}{dx}\right|_{x=1} = 3(1)^2 - 6(1) + 6 $$ $$ \text{Slope} = 3 - 6 + 6 $$ $$ \text{Slope} = 3 $$