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Expert: Alon Mandes - 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
Ok Melissa, you got the derivative right. To find critical points
we must set f'(x)=0. That means -2xe^(-x^2)=0, we can see that the
f'(x)=0 only in x=0, & x->∞.
For x<0 then f'(x)>0. f(x) increasing.
For x>0 then f'(x)<0. f(x) decreasing.
Let's calculate the 2nd derivative , in order to determine whether
x=0 is max or min :
f''(x)=[-2xe^-x^2]'=-2e^(-x^2)+4x^2e^(-x^2).
f''(0)=-2e^0+4*0*e^(0)=-2 <0 which means max.
Let's now, substitute the 3 points 0,-2,1 to determine the absolute
min/max :
f(0)=e^0=1
f(-2)=e^-4
f(1)=e^(-1)=1/e
So, in the interval [-2,1] : x=0 is the global maximum & x=-2 is the
global minimum.

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

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M.A in Mathematics & Bs.c in Electronics.

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