Step 1: Understanding the Concept:
The "slope of a curve" at a specific point is equivalent to the value of the first derivative of the function at that point.
Geometrically, the derivative \( \frac{dy}{dx} \) represents the gradient of the tangent line drawn to the curve at the point \( (x, y) \).
To find the slope at \( x = 2 \), we must first differentiate the function and then evaluate that derivative at the given value.
Step 2: Key Formula or Approach:
The power rule for differentiation states that for any term \( x^n \), the derivative is \( n \cdot x^{n-1} \).
For a function \( y = f(x) \), the slope at \( x = a \) is \( f'(a) \).
Step 3: Detailed Explanation:
The given equation of the curve is:
\[ y = x^3 - 3x^2 + 2 \]
Differentiate both sides with respect to \( x \):
Using the power rule term-by-term:
- The derivative of \( x^3 \) is \( 3x^2 \).
- The derivative of \( -3x^2 \) is \( -3(2x) = -6x \).
- The derivative of the constant \( 2 \) is \( 0 \).
Thus, the derivative function is:
\[ \frac{dy}{dx} = 3x^2 - 6x \]
This function gives the slope at any general point \( x \).
To find the specific slope at the point where \( x = 2 \), substitute \( 2 \) into the derivative:
\[ \text{Slope} = \left. \frac{dy}{dx} \right|_{x=2} \]
\[ \text{Slope} = 3(2)^2 - 6(2) \]
\[ \text{Slope} = 3(4) - 12 \]
\[ \text{Slope} = 12 - 12 \]
\[ \text{Slope} = 0 \]
Step 4: Final Answer:
The slope of the curve at \( x = 2 \) is \( 0 \).
This indicates that the tangent line is horizontal at this point (which is a critical point).
The correct option is (A).