Question

Application of Derivatives

Problem ....

Find the intervals where the function \(f(x) = x^2 e^x\) is increasing and decreasing.

Answer 1

@ganeshie8

and @BSwan

This is what I have done yet.

From the last problem as I differentiated it first, so, I will also differentiate this one.

f'(x) = \(2xe^x + x^2 e^x\)

Answer 2

good , then ?

Answer 3

\(f'(x) = x e^x ( x + 2) \) ?

Answer 4

For increasing , it is true that it should be \(\ge\) 0 and for decreasing it should be \(\le 0\) , right?

Answer 5

ahaaa :D

Answer 6

So, making conditions :

For \(\bf{Increasing}\) :

\(\boxed{\mathsf{xe^x (x+2) \ge 0 }}\)

For \(\bf{Decreasing}\) :

\(\boxed{\mathsf{xe^x(x+2) \le 0}}\)

Answer 7

I'm not sure how to go on from here. Though, it should be easy to continue from this point, but am not sure how to do that.

Answer 8

e^x is always positive

Answer 9

Oh... that may help.

Answer 10

So, x can be lesser than -2 ... and x can be greater than 0 .. right?

Answer 11

for increasing.

Answer 12

x(x+2) >=0
when x <-2 or x>0

x(x+2) <= 0
when x>-2 and x< 0

Answer 13

\(x \in (-\infty , -2) \cup (0 , \infty)\)

Answer 14

That was for increasing^

And for decreasing, I get :

\( x \in (-2,0) \)

Answer 15

o.O no...............!

Answer 16

pick up any points in the intervals to see if it increas or decrease
with f'(x) >0 test

Answer 17

For increasing...

\(x \in (-\infty , 2] \cup [ 0 ,\infty)\)

and for decreasing..

\(x \in [-2,0] \)

Is this right?

Answer 18

Looks good !

Answer 19

I'm confused about the closed and open brackets. What the book (course) , says is :

f(x) is increasing in \([-\infty , -2) \cup [0, \infty)\)

f(x) is decreasing in \((-2,0]\)

Answer 20

it is illegal to include \([-\infty \)

Answer 21

Yeah... Moreover, -2 should be included in increasing as well as in decreasing.

Answer 22

^Agree !

Answer 23

i dnt dnt like the \(\cup\) notation xD

Answer 24

Probably, just a printing mistake.

As this is what it says.

@BSwan -> I used to hate Cup and cap notations, but, now when I get them, I love them!

Answer 25

haha @BSwan please medal @mathslover :)

Answer 26

well its not like that , previously i state why its not true to put them , so they still teach this in pre cal although it make a lote of doubts !

Answer 27

:-) Okay, I have got some more question. I should close it now.

Thanks a lot @BSwan and @ganeshie8 for being there to help me out everytime. I really thank you both for that.

Answer 28

np :)