Final answer:
To find the equation of the line that is tangent to both y = 1/x and y = -x², set the two equations equal to each other and solve for the x-values of the points of intersection. Use the slope formula and point-slope form to find the equation of the line.
Step-by-step explanation:
To find the equation of the line that is tangent to both y = 1/x and y = -x², we need to find the points where the two curves intersect. To do this, set the two equations equal to each other:
1/x = -x²
Solve this equation to find the x-values of the points of intersection.
Once you have the x-values, substitute them back into either equation to find the corresponding y-values.
Now that you have two points on the line, you can use the slope formula to find the slope of the line:
m = (y₂ - y₁) / (x₂ - x₁)
Finally, use the point-slope form of a linear equation to find the equation of the line:
y - y₁ = m(x - x₁)