Derivatives in relation to curve sketching

[scorpa]

Hi Again!

Right now I'm taking derivatives in relation to curve sketching, and I just wanted to make sure I am doing these right.

The first question is to determine using the first derivative where the graph of y = x is rising.
(x^2)+1

This is what I have done so far:

I tried to find the derivative using the quotient rule...

((x^2)+1)(d/dx)(x)-(x)(d/dx)((x^2)+1)
((x^2)+1)^2


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

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

That is the value I found for the first derivative, although I am unsure whether I have done it right or not. Then to find where the curve was rising I said that since the derivative must be greater than 0, the value I found for the derivative must be greater than zero. Then I tried solving that and I ended up with -x^2 + 1 > x^4 + 2x^2 + 2 and although I realize that that is not quite finished yet, it just doesn't seem right to me. I am obviously going wrong somewhere, but where I do not know.


The second question asks you to determine where f(x) = x^2(1-x) is concave upward using the second derivative. This is what I did:

(x^2)(d/dx)(1-x) + (1-x)(d/dx)(x^2) =


(x^2)(-1) + (1-x)(2x) =

(-3x^2) + 2x This is the first derivative I found using the product rule and the steps shown above.

Then I found the second derivative to be -6x + 2

Have I found the derivatives correctly for these questions? It just seems like something is very wrong, mostly with the first question. Any help you guys can give me I would really appreciate. Thanks in advance.

 

[dextercioby]

a) The first derivative is correct.However,you solved this inequation

$$\frac{-x^{2}+1}{\left(x^{2}+1\right)^{2}} >0$$

incorrectly...

Do it again HINT:The denominator is always positive

 

[dextercioby]

For the second,everything is correct so far.How do you interpret the result...?

Daniel.

 

[scorpa]

Thank you so much for the help, I really appreciate it!

Ok for the concave one ( second question) I think it is concave upward from (-infinity, (1/3))

I solved it like this:

-6x+2>0
-6x>-2
x < (1/3)

I'm redoing the last part of the first question right now, I post again in a minute.

 

[dextercioby]

Okay.It's okay,so far.

Daniel.

 

[scorpa]

OK now for the first one,

I went back and did this:

(-x^2)+1 >0
((x^2)+1)^2

=

x^2 < 1

x<1
x>-1

therefore -1 < x < 1


OK now an I on the right track? Thanks again for the help!

 

[dextercioby]

It's perfect,u can plot it to get a graphical confirmation,but it's everything okay now.

Daniel.

 

[scorpa]

Alright! Thank you so much!