Find all values of x for which the tangent line ot y=2x^3-x^2 is perpendicular to the line x +4y=10?
To find the values of x for which the tangent line to the equation y = 2x^3 - x^2 is perpendicular to the line x + 4y = 10, we need to follow these steps:
Step 1: Find the slope of the given line.
- The equation of the given line is x + 4y = 10, we can rewrite it in slope-intercept form as y = -1/4x + 5/2.
- The slope of the given line is -1/4.
Step 2: Find the derivative of the equation y = 2x^3 - x^2.
- Take the derivative of y with respect to x using the power rule.
- The derivative of 2x^3 is 6x^2, and the derivative of -x^2 is -2x.
- Thus, the derivative of y = 2x^3 - x^2 is dy/dx = 6x^2 - 2x.
Step 3: Find the slope of the tangent line.
- Since the derivative gives us the slope of the tangent line, the slope of the tangent line y = 2x^3 - x^2 is given by dy/dx = 6x^2 - 2x.
Step 4: Determine the condition for perpendicular lines.
- Two lines are perpendicular to each other if and only if the product of their slopes is equal to -1.
Step 5: Set up the equation to find the values of x.
- To find the values of x, we need to set up the equation for the product of the slopes.
- The slope of the given line is -1/4, and the slope of the tangent line is 6x^2 - 2x.
- So, we have (-1/4)(6x^2 - 2x) = -1.
Step 6: Solve the equation to find the values of x.
- Distribute -1/4 to 6x^2 - 2x: -6/4x^2 + 2/4x = -1.
- Simplify the equation: -3/2x^2 + 1/2x = -1.
- Multiply through by 2 to clear the fractions: -3x^2 + x = -2.
- Rearrange the equation to standard form: 3x^2 - x - 2 = 0.
- Factorize the quadratic equation: (3x + 2)(x - 1) = 0.
- Set each factor equal to zero and solve for x:
3x + 2 = 0 --> x = -2/3,
x - 1 = 0 --> x = 1.
Step 7: Verify the solutions.
- The solutions for x are x = -2/3 and x = 1.
- Substitute these values into the derivative dy/dx = 6x^2 - 2x and check if the slopes of the tangent lines are perpendicular to the given line.
Therefore, the tangent line to the equation y = 2x^3 - x^2 is perpendicular to the line x + 4y = 10 at x = -2/3 and x = 1.