Final answer:
The line (y = 3x + k) is tangent to the curve (y = x³) when k is equal to -1 or 1.
Step-by-step explanation:
The line (y = 3x + k) is tangent to the curve (y = x³) when k is equal to -1 or 1.
To find the value of k that makes the line tangent to the curve, we need to set the derivatives of the line and the curve equal to each other.
The derivative of the line (y = 3x + k) is 3, and the derivative of the curve (y = x³) is 3x².
Setting these derivatives equal to each other gives us 3 = 3x². Solving for x, we get x = ±1. Substituting x = 1 into the line equation gives us y = 3(1) + k = 3 + k, and substituting x = -1 gives us y = 3(-1) + k = -3 + k.
Therefore, when x = 1, y = 3 + k, and when x = -1, y = -3 + k. In order for the line to be tangent to the curve, the y-values of the line and the curve need to be equal at the point of tangency. This means that 3 + k = 1³ and -3 + k = (-1)³. Simplifying these equations, we get 3 + k = 1 and -3 + k = -1. Solving for k gives us k = -1 or k = 1.