Determine the equations for the tangents to the points on the curve y=-x^4 + 8x^2 such that the tangent lines are perpendicular to the line x=1

Question

Determine the equations for the tangents to the points on the curve y=-x^4 + 8x^2 such that the tangent lines are perpendicular to the line x=1

Answer 1

tangent lines perpendicular to x = 1 have zero slope.

 

y = -x^4 + 8x^2

 

y' = -4x^3 + 16x = -4x(x^2 - 4) = -4x(x + 2)(x - 2)

 

y' = 0 when x = -2, 0, +2

 

y(±2) = -16 + 8(4) = 16

y(0) = 0

 

y - 16 = 0

 

y = 16 is tangent when x = ±2.

 

y = 0 is tangent when x = 0.