Expert: Paul Klarreich - 11/29/2008

Question
Given the function f(x)=e^-x^2

(a). Find the derivative of f
(b). Find the critical values
(c). Draw a sign diagram that shows the critical values and where the derivative is positive and negative.
(d). Find the relative maxima and relative minima (if any).
(e). Find the intervals where the function is increasing and where it is decreasing.
(f). Find the absolute minimum and absolute maximum on the interval [-2, 1].

***Ok i really need help with this. i think the derivative is -2xe^-x^2 but i didnt think it had any critical values....Can u please help me!?!?


Answer
Questioner:   melissa
Category:  Calculus
Private:  No
 
Subject:  Calculus question
Question:  Given the function f(x)=e^-x^2

(a). Find the derivative of f
(b). Find the critical values
(c). Draw a sign diagram that shows the critical values and where the derivative is positive and negative.
(d). Find the relative maxima and relative minima (if any).
(e). Find the intervals where the function is increasing and where it is decreasing.
(f). Find the absolute minimum and absolute maximum on the interval [-2, 1].

***Ok i really need help with this. i think the derivative is -2xe^-x^2 but i didnt think it had any critical values....Can u please help me!?!?
......................................................
Hi, Melissa,

(a). Find the derivative of f

Your derivative is correct -- f'(x) = - 2x e^(-x^2)

Now there is a nice thing about  e^(whatever) -- it is never negative and it is never zero.  That means if you are doing anything involving questions like : is it negative? is it zero? is it positive?, then you can safely ignore this factor.

You can focus on: where is (-2x) zero? positive? negative?   Because:

The sign of  (-2x)*(something that is always positive) is
the same as the sign of  (-2x)
....................
(b). Find the critical values

Have you written out and studied this vocabulary term?

Obviously  x = 0.  (Why?)
.........................
(c). Draw a sign diagram that shows the critical values and where the derivative is positive and negative.

Now just test:  

Where is  -2x > 0  (positive?)  Obviously  x < 0. (Why?)
Where is  -2x < 0  (negative?)  Obviously  x > 0. (Why?)

Draw your diagram from that.

.............................
(d). Find the relative maxima and relative minima (if any).

Isn't that the same as (b)?  Oh, yes, you are expected to actually get the y-values, so do that.

................................
(e). Find the intervals where the function is increasing and where it is decreasing.

Isn't that the same as (you guess which part)?
.................................
(f). Find the absolute minimum and absolute maximum on the interval [-2, 1].

Now do some logical, er.. deductive reasoning:

If f(x) is increasing up to  x = 0 and decreasing after that, what do you think about the value at  x = 0?