The line \(y = mx - 3\) is a tangent to the curve \(y = 1 - 3x + 2x^{3}\) at (1, 0). Find the value of the constant m.
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Correct Answer: Option n
Explanation:
\(y = 1 - 3x + 2x^{3}\)
\(\frac{\mathrm d y}{\mathrm d x} = -3 + 6x^{2}\)
At (1, 0), \(\frac{\mathrm d y}{\mathrm d x} = -3 + 6(1^{2}) = -3 + 6 = 3\)
\(y = mx - 3 \implies \frac{\mathrm d y}{\mathrm d x} = m = 3\) (Tangent with equal gradient)
\(y = 1 - 3x + 2x^{3}\)
\(\frac{\mathrm d y}{\mathrm d x} = -3 + 6x^{2}\)
At (1, 0), \(\frac{\mathrm d y}{\mathrm d x} = -3 + 6(1^{2}) = -3 + 6 = 3\)
\(y = mx - 3 \implies \frac{\mathrm d y}{\mathrm d x} = m = 3\) (Tangent with equal gradient)