The tangent to the curve \(y = 4x^{3} + kx^{2} - 6x + 4\) at the point P(1, m) is parallel to the x- axis, where k and m are constants. Determine the coordinates of P.
A. (1, 2) B. (1, 1) C. (1, -1) D. (1, -2)
Correct Answer: C
Explanation
\(y = 4x^{3} + kx^{2} - 6x + 4\) \(\frac{\mathrm d y}{\mathrm d x} = 12x^{2} + 2kx - 6\) At P(1, m) \(\frac{\mathrm d y}{\mathrm d x} = 12 + 2k - 6 = 0\) (parallel to the x- axis) \(6 + 2k = 0 \implies k = -3\) \(P(1, m) \implies m = 4(1^{3}) - 3(1^{2}) - 6(1) + 4) = -1 P = (1, -1)
