Two different tangents which are perpendicular?

In summary, the question is whether the curves y=x^2 and y=x^3 have two different tangents that are perpendicular, and if so, how to prove it. This can be shown by finding two points, x1 and x2, on each curve where the product of their slopes is -1. For the quadratic curve, this would be (2x1)(2x2)=-1, and for the cubic curve, it would be (3x12)(3x22)=-1.
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Homework Statement


Does the curve y=x^2 have two different tangents which are perpendicular? Does the curve y=x^3?

The Attempt at a Solution


I have no idea how to prove that for x^2 or x^3.
(x^2)' =2x that is tangent at a point x
then m = -1/2 x for another point...
How do i prove that there is sucha point? then how about x^3?
 
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  • #2


Two lines are perpendicular if and only if the product of their slopes is -1. So the question is, "are the two different values of x, say x1 and x2, such that (2x1)(2x2)= -1?"
For the cubic, the corresponding equation is (3x12)(3x22)= -1.
 

1. What does it mean for two tangents to be perpendicular?

Two tangents are said to be perpendicular when they intersect at a right angle.

2. How can you determine if two tangents are perpendicular?

You can determine if two tangents are perpendicular by checking if the slopes of the tangents are negative reciprocals of each other. If they are, then they are perpendicular.

3. Can two tangents be perpendicular at more than one point?

Yes, two tangents can be perpendicular at more than one point. This happens when the curves they are tangent to intersect at multiple points.

4. What is the significance of two tangents being perpendicular?

When two tangents are perpendicular, it indicates that the curves they are tangent to are changing direction at that point.

5. Are perpendicular tangents exclusive to circles?

No, perpendicular tangents can exist for any type of curve. However, they are most commonly associated with circles due to their symmetrical nature.

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