# Thread: show that a tangent line to a curve is tangent at another point

1. ## show that a tangent line to a curve is tangent at another point

Show that the tangent line to the curve y = (x^2 + x - 2)^3 + 3 at (1, 3) is also tangent to the curve at another point.

Like my previous question I got the derivative

being 3(2x+1)(x^2 +x -2)^2

I am not sure where to go from here.

Thank you!

2. ## Re: show that a tangent line to a curve is tangent at another point

What is the gradient at that point?

3. ## Re: show that a tangent line to a curve is tangent at another point

solve dy/dx for x=1 I get a slope of 18

4. ## Re: show that a tangent line to a curve is tangent at another point

No, if you let x = 1 in dy/dx you should get 0.

5. ## Re: show that a tangent line to a curve is tangent at another point

Originally Posted by Jonroberts74
solve dy/dx for x=1 I get a slope of 18
Nope. Your derivative is correct, but you computed the slope incorrectly. At x = 1:
$y' = 3(2(1) + 1)((1)^2 + (1) - 2)^2 = ?$

Now set your derivative equal to 0. What values of x are there where y' = 0?

-Dan

Gah! ProveIt beat me to it.

6. ## Re: show that a tangent line to a curve is tangent at another point

x= -1/2, x = 1, x = -2

need more work on my factoring skills.

thank you!

7. ## Re: show that a tangent line to a curve is tangent at another point

To find the other point that the tangent line touches, note that
$\frac{dy}{dx}=0 \text{ and also, your tangent line is given by } y=3, \text{ so set } y=3 \implies 3=(x^2+x-2)^3 + 3 \implies 0=(x^2+x-2)^3 \implies 0=(x^2+x-2) \implies \cdots \text{ (you can practice your factoring here! =) )}$

Just a note, in this question the tangent line touches 2 local extrema at the same level (i.e. y=3), but not necessarily.

8. ## Re: show that a tangent line to a curve is tangent at another point

Originally Posted by chen09
Just a note, in this question the tangent line touches 2 local extrema at the same level (i.e. y=3), but not necessarily.
Actually the points (-2, 3) and (1, 3) are inflection points.

-Dan

9. ## Re: show that a tangent line to a curve is tangent at another point

Thanks, I didn't check that. I was rushing, and got 2 problem sets due Monday PDE + topology =(

I apologize for being emo in this thread.

10. ## Re: show that a tangent line to a curve is tangent at another point

dang, I hope someday to understand PDEs and topology

11. ## Re: show that a tangent line to a curve is tangent at another point

Originally Posted by Jonroberts74
dang, I hope someday to understand PDEs and topology
Me too!

-Dan