Where on the curve y = (4+x^2)^-1 does the tangent line have the greatest slope?

Question

tkhunny · Answer

Have you considered the 1st Derivative to create a function describing the slope at each point??

Have you considered the 2nd derivative to find the extrema of this new function?

anonymous · Answer

Yes. I took each derivative and got the first to be -2x(4+x^2)^-2 and the second to be (6x^2 - 8) / (4+x^2)^3

anonymous · Answer

I thought I was to make the second derivative equal to zero, find the points, determine which is a max, and plug it into the derivative to determine the y value for the determined x value.

tkhunny · Answer

Okay, then why not just answer the question?

anonymous · Answer

I did, but apparently I keep getting it wrong.

tkhunny · Answer

Why do you think your answer is incorrect?

anonymous · Answer

The online program says its wrong

anonymous · Answer

We use WileyPLUS

tkhunny · Answer

Okay.  Your derivatives look good.  Where else could it go wrong?

anonymous · Answer

So I get that x = $$\sqrt{4/3}$$, yeah?

tkhunny · Answer

That's good, as long as you get both + and -.  Which is the greater slope?

anonymous · Answer

Blah, it's the negative root. >_< I probably kept forgetting the sign

tkhunny · Answer

Very good.  Now, we have $x = -\dfrac{2}{3}\sqrt{3}$.  How do we find y?

anonymous · Answer

Why is x = to -2/3 sqrt3?

tkhunny · Answer

Standard simplification.  $x = -\dfrac{2}{\sqrt{3}} = -\sqrt{\dfrac{4}{3}}$ if you like.

anonymous · Answer

Fair enough, I am quite terrible at roots

tkhunny · Answer

Gotta get up to speed on that!

anonymous · Answer

Yeah, that and a lot of things. >_< Alright, y is..

anonymous · Answer

Wait, so was the -2/3 a typo or am I missing another part?

tkhunny · Answer

The moment of truth.  In which or our three functions will you substitute our x-value to determine the proper y-value?

anonymous · Answer

First derivative, I want to find out when it is greatest

tkhunny · Answer

There where we wandered off!  The question is "where on the curve is the greatest slope", not "what is the greatest slope".  Please substitute into the original function to obtain the answer to the problem statement.

anonymous · Answer

sigh. This chapter...

tkhunny · Answer

Hang in there.  You are doing well.  Just pay a little better attention.

anonymous · Answer

3/16?

tkhunny · Answer

That's it.  See, good work!

anonymous · Answer

Thanks a lot for the help, its definitely much appreciated!

tkhunny · Answer

No worries.  We're here to help.  It's a great pleasure to see when it does help.