1. ## increasing or decreasing

the question is

Let F(a) be the function which gives the area under the graph of y=x*e^-x between x=0 and x=a for a>0

i found F(a) to be (-x*e^-x)-(e^-x)
then evaluated with the given endpoints i got

(-a*e^-a)-e^a+1
(not sure if this is correct)

the second question is

is F increasing or a decreasing function?
is it the same that if the second derivative tells the concavity? pointers anyone?

2. Originally Posted by acosta0809
is F increasing or a decreasing function?
is it the same that if the second derivative tells the concavity? pointers anyone?
Look at it like any other function.

$\displaystyle \frac{dF}{dx}=x*e^{-x}\Rightarrow\frac{d}{dx}(x*e^{-x})=e^{-x}-xe^{-x}=0$

$\displaystyle \frac{1}{e^x}(1-x)=0\Rightarrow\text{ critical values at }x=1$

3. Hello, acosta0809!

Let $\displaystyle F(a)$ be the function which gives the area under the graph of $\displaystyle y\,=\,xe^{-x}$
between $\displaystyle x=0$ and $\displaystyle x=a$ for $\displaystyle a>0.$

i found $\displaystyle F(a)$ to be: .$\displaystyle -xe^{-x} -e^{-x}$

then evaluated with the given endpoints i got: .$\displaystyle -ae^{-a} -e^{-a}+1$
(not sure if this is correct)
It's correct!

The second question is:

Is $\displaystyle F(a)$ an increasing or a decreasing function?
Find the first derivative (slope).

We have: .$\displaystyle F(a) \:=\:ae^{-a} - e^{-a} + 1$

Then: .$\displaystyle F'(a) \;=\;ae^{-a} - e^{-} + e^{-a} + 0 \;=\;\frac{a}{e^a}$

Since $\displaystyle a > 0$, then: .$\displaystyle F'(a) \:=\:\frac{a}{e^a}$ is always positive.

Therefore, $\displaystyle F(a)$ is an increasing function.

If we look at the graph, it's obvious . . .
Code:
. . . - |
. . . - |     ..*.
. . . - |   *::::::*.
. . . - | *::::::::::*..
. . . - |*::::::::::::::::*
. . . - |:::::::::::::::::|        *
. . . - * - - - - - - - - + - - - - -
. . . - |                 a
As $\displaystyle a$ increases, the area under the curve increases.