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Explain Bot To find the equation of the line(s) tangent to the curve y = x^3 - 6x + 2 and parallel to the line y = 6x - 2, we need to take the following steps:
1. Find the derivative of the curve y = x^3 - 6x + 2. This will give us the slope of the curve at any point.
2. Set the derivative equal to the slope of the parallel line, which is 6 in this case, since parallel lines have the same slope.
3. Solve the equation to find the x-values where the slopes are equal. These x-values will be the x-coordinates of the points where the tangent lines touch the curve.
4. Substitute the x-values obtained in step 3 into the original equation y = x^3 - 6x + 2 to find the corresponding y-values.
5. Use the point-slope form of the equation of a line, y - y₁ = m(x - x₁), where (x₁, y₁) is the point of tangency and m is the slope of the tangent line, to write the equation(s) of the tangent line(s).
Let's go through these steps one by one:
Step 1: Find the derivative of the curve y = x^3 - 6x + 2.
To find the derivative, we differentiate the equation with respect to x:
dy/dx = 3x^2 - 6.
Step 2: Set the derivative equal to the slope of the parallel line.
Since the parallel line y = 6x - 2 has a slope of 6, we set the derivative equal to 6:
3x^2 - 6 = 6.
Step 3: Solve the equation to find the x-values.
Rearrange the equation:
3x^2 - 6 - 6 = 0,
3x^2 - 12 = 0.
Simplify further:
x^2 - 4 = 0.
Factor the equation:
(x - 2)(x + 2) = 0.
Solve for x:
x - 2 = 0 or x + 2 = 0.
x = 2 or x = -2.
Step 4: Find the corresponding y-values.
Substitute the x-values into the original equation y = x^3 - 6x + 2:
For x = 2:
y = (2)^3 - 6(2) + 2 = 8 - 12 + 2 = -2.
For x = -2:
y = (-2)^3 - 6(-2) + 2 = -8 + 12 + 2 = 6.
Step 5: Write the equation(s) of the tangent line(s).
Using the point-slope form, we can write the equations as follows:
For the tangent line at (2, -2):
y - (-2) = 6(x - 2).
Simplifying this equation, we get:
y + 2 = 6x - 12.
For the tangent line at (-2, 6):
y - 6 = 6(x + 2).
Simplifying this equation, we get:
y - 6 = 6x + 12.
Therefore, the equations of the lines tangent to the curve y = x^3 - 6x + 2 and parallel to y = 6x - 2 are:
y + 2 = 6x - 12 and y - 6 = 6x + 12.