Hi, im having problems with the following problem. The main issue is actually starting the problem.
Find the two points on the curve y = x^4 - 2x^2 - x that have a common tangent line.
First, find the derivative of y(x) so that you know the slope of the tangent line at any given x.
The derivative will be a quadratic formula for which there may be two solutions.
To find the derivative of the function y(x) = x^4 - 2x^2 - x, you can apply the power rule.
The power rule states that for any term of the form x^n, the derivative is given by nx^(n-1).
Applying the power rule to each term in the function, we get:
dy/dx = 4x^3 - 4x - 1
Now that we have the derivative, we can find the slope of the tangent line at any given x.
Next, let's set the derivative equal to the same slope value and solve for x to find the x-values of the points where the tangent line is common.
Set dy/dx = m, where m is the slope of the tangent line.
So, we have:
4x^3 - 4x - 1 = m
This is a cubic equation, and in general, it can have up to three roots. We need to find the values of x for which this equation holds true for the given slope m.
To solve this equation, you can use numerical methods such as Newton's method or graphing calculators to approximate the roots.
By finding all possible values of x, you can substitute them back into the original function y(x) to find the corresponding y-values. These will be the two points on the curve that have a common tangent line.