Find if function is increasing or decreasing (Find the derivative)

$\displaystyle ne^{-\frac{n^2}{8}}$

$\displaystyle ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}} $

with quotient rule

$\displaystyle \frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}$

$\displaystyle = \frac{(e^{\frac{n^2}{8}})-(n)(\frac{n}{4})}{(e^{\frac{n^2}{8}})^2}$

$\displaystyle = \frac{1 - \frac{n^2}{4}}{e^\frac{n^2}{8}}$

What can I do next? Or can I assume the function is decreasing now?

Re: Find if function is increasing or decreasing (Find the derivative)

Quote:

Originally Posted by

**moonman** $\displaystyle ne^{-\frac{n^2}{8}}$

$\displaystyle ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}} $

with quotient rule

$\displaystyle \frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}$

$\displaystyle = \frac{(e^{\frac{n^2}{8}})-(n)(\frac{n}{4})}{(e^{\frac{n^2}{8}})^2}$

$\displaystyle = \frac{1 - \frac{n^2}{4}}{e^\frac{n^2}{8}}$

What can I do next? Or can I assume the function is decreasing now?

... what if $\displaystyle |n| < 2$ ?

Re: Find if function is increasing or decreasing (Find the derivative)

Quote:

Originally Posted by

**skeeter** ... what if $\displaystyle |n| < 2$ ?

If $\displaystyle |n| < 2$ then the function would be positive...so then then the function is positive?

Re: Find if function is increasing or decreasing (Find the derivative)

no ... what does $\displaystyle f' > 0$ tell you about $\displaystyle f$ ?

Re: Find if function is increasing or decreasing (Find the derivative)

Quote:

Originally Posted by

**skeeter** no ... what does $\displaystyle f' > 0$ tell you about $\displaystyle f$ ?

That tells me that f is increasing?

Re: Find if function is increasing or decreasing (Find the derivative)

The derivative of the function vanishes at n = 2, -2? This function cannot be monotonic. Looks to me that it has points of extrema. It can only be locally increasing or decreasing.

Re: Find if function is increasing or decreasing (Find the derivative)

So can the Integral Test be performed on this if it were a series?

Re: Find if function is increasing or decreasing (Find the derivative)

Ok, I found an error in differentiation.

Re: Find if function is increasing or decreasing (Find the derivative)

$\displaystyle ne^{-\frac{n^2}{8}}$

$\displaystyle ne^{-\frac{n^2}{8}} = \frac{n}{e^\frac{n^2}{8}} $

with quotient rule

$\displaystyle \frac{n}{e^\frac{n^2}{8}}= \frac{(n)'(e^{\frac{n^2}{8}})-(n)(e^{\frac{n^2}{8}})'}{(e^{\frac{n^2}{8}})^2}$

$\displaystyle = \frac{(e^{\frac{n^2}{8}})-(n)(e^\frac{n^2}{8})(\frac{n}{4})}{(e^{\frac{n^2}{ 8}})^2}$

$\displaystyle =\frac{1-\frac{n^2}{4}}{1}$ always negative

Correct? Can someone please check the derivative?

Re: Find if function is increasing or decreasing (Find the derivative)

you had it correct the first time ...

$\displaystyle \frac{d}{dx} \left[x \cdot e^{-\frac{x^2}{8}}\right] $

$\displaystyle x\left(-\frac{x}{4}\right)e^{-\frac{x^2}{8}} + e^{-\frac{x^2}{8}}$

$\displaystyle e^{-\frac{x^2}{8}}\left[1 - \frac{x^2}{4}\right]$

the derivative is negative for $\displaystyle |x| > 2$ and positive for $\displaystyle |x| < 2$