Question

The slope of the tangent to the curve \( y = 3x^2 - 5x + 6 \) at \( (1,4) \) is

• A. \( -2 \)
• B. \( 1 \)
• C. \( 0 \)
• D. \( -1 \)
• E. \( 2 \)

Hint:

Always verify the point lies on the curve before finding slope.

Answer 1

The Correct Option is B

Concept: The slope of the tangent to a curve at any point is given by the derivative of the function at that point. \[ \text{Slope of tangent} = \frac{dy}{dx} \]

Step 1: Verify that the point lies on the curve.
Given curve: \[ y = 3x^2 - 5x + 6 \] Substitute \( x = 1 \): \[ y = 3(1)^2 - 5(1) + 6 \] \[ = 3 - 5 + 6 = 4 \] Thus, point \( (1,4) \) lies on the curve.

Step 2: Differentiate the function.
\[ y = 3x^2 - 5x + 6 \] Differentiate term by term: \[ \frac{d}{dx}(3x^2) = 6x \] \[ \frac{d}{dx}(-5x) = -5 \] \[ \frac{d}{dx}(6) = 0 \] Thus: \[ \frac{dy}{dx} = 6x - 5 \]

Step 3: Find slope at \( x = 1 \).
\[ \frac{dy}{dx} = 6(1) - 5 \] \[ = 6 - 5 = 1 \]

Step 4: Interpretation.
This means:
• The tangent line at \( (1,4) \) has slope \(1\)
• The curve is increasing at this point

Step 5: Final Answer.
\[ \boxed{1} \]