Question

e^(-x^6) absolute maximum and minimum value

Answer 1

Hey Rina :) Welcome to OpenStudy!

Are we using Calculus techniques to figure this out?
Or is this for an earlier class? Algebra or something? :)

Answer 2

calculus

Answer 3

\[\Large\rm y=e^{-x^6}\]To find max/min we'll first need to find critical points and then classify those points further.
Critical points exist where the first derivative is zero.
Remember how to differentiate an exponential?\[\Large\rm \frac{d}{dx}e^{stuff}=e^{stuff}\frac{d}{dx}(stuff)\]We get the `same exponential back` but we have to apply the `chain rule`, multiplying by the derivative of the stuff in the exponent.

Answer 4

\[\Large\rm y'=e^{-x^6}(-x^6)'\]\[\Large\rm y'=e^{-x^6}(-6x^5)\]That part ok?

Answer 5

Then we want to look for critical points,\[\Large\rm 0=e^{-x^6}(-6x^5)\]

Answer 6

ok

Answer 7

We'll apply the `Zero-Factor Property`, setting each individual factor equal to zero and solving for x in each case.\[\Large\rm 0=e^{-x^6},\qquad\qquad\qquad 0=(-6x^5)\]In the first case here, the exponential function is`never` zero.
So we're not getting any critical points from this part:\[\Large\rm \cancel{0=e^{-x^6}}\]

Answer 8

How bout the other portion? Does that give us any critical points?

Answer 9

yes I think it would be x=0

Answer 10

Ok great!
So we've found our critical point.
Now we need to find out what type of critical point it is.

Answer 11

ok

Answer 12

Oh I forgot to mention that they also gave an interval which was -3≤x≤1

Answer 13

Ok since we're given an interval, that makes things a little easier.
We must also involve the end points.

So what we'll do is plug our critical points, and end points into the original function.
The largest output is our maximum,
while the smallest output is our minimum.

Answer 14

so I have to plug the critical point and the given interval into the original equation to find the abs max and min

Answer 15

oh oh ok

Answer 16

\[\Large\rm y(x)=e^{-x^6}\]
\[\Large\rm y(-3)=?\]\[\Large\rm y(0)=?\]\[\Large\rm y(1)=?\]

Answer 17

y(-3)=0, y(0)=1 , y(1)=.37

Answer 18

right?

Answer 19

Mmm yes very good.
So do you understand how to pick out your max and min from those three outputs?

Answer 20

yes and thank you sir

Answer 21

List your max/min as ordered pairs.
It might be a better idea to leave your y(-3) in exact form.
Yes, it's very very close to zero, but it's not 0.\[\Large\rm y(-3)=e^{-729}\]It's something like that^
Just depends what your teacher prefers I guess.

Answer 22

ok

Answer 23

I will leave it that form and thankx again