determine the values of x such that y=x^2 and y=1/x have the same slope

Answer 1

y = x^2

y' = 2x

y = 1/x = x^-1

y' = -1/x^2

set them equal

2x = -1/x^2

x^3 = -1/2

x = cube root of -1/2 = -1/2^(1/3) = about - 0.7937

-.7937^3 =about -.499999006= about -1/2

x^2 is an upward opening parabola with vertex at the origin

1/x^2 is graphically 2 branches of a rectangular hyperbola with the right branch reflected over the x axis

graph each and you can see how the slopes are equal in about where x=-.7937

but the slopes that are equal are in different quadrants,

Answer 2

2x = - 1/x²; 2x³ = - 1; x = (-1/2)^1/3