Let f(x)=xe^x A)Determine domain and range? B)Is f(x) an odd function, an even f

Question

Let f(x)=xe^x

A)Determine domain and range?

B)Is f(x) an odd function, an even function ,or neither? Explain your answer using APPROPRIATE TESTS!

C)Find the roots of f(x). DETERMINE where f(x) is positive and where f(x) is negative?

D)Evaluate lim(x->-inifinity) f(x) using L'Hopital's Rule?

E)Determine if there are any asymptotes by calculating the appropriate limits. State the equation(s) of any asymptotes.

F)CALCULATE f'(x) and f''(x).Write your answers in factored form.

G)WHAT is A CRITICAL NUMBER for f(x)? Determine the CRITICAL numbers for f(x)

H)Use the critical numbers you found in part (g) to divide the REAL line INTO INTERVALS. dETERMINE WHERE f(x) is increasing or decreasing by analysing the sign of f'(x) inside EACH of these intervals.

I)TEST THE CRITICAL NUMBERS AND IDENTIFY ANY maxima and minina of f(x). Justif your answer. Calculate the Y-COMPONENT OF each maximum or minimum .

J) Determine concavity of f(x), up or down, explain your answer!FIND ANY POINTS OF INFLECTION FOR f(x).DETERMINE BOTH X AND Y COORDINATES. LEAVE YOUR ANSWER IN SYMBOLIC FORM

k) SKETCH THE GRAPH WITH ALL IMPORTANT POINTS AND LABELLING

Explanation / Answer

f(x)=xe^x

here x can take any value so domain is ( -inf , inf )

f(-x) = -xe^-x

-f(x) = -xe^x

so f(x) not equal to -f(x) and f(-x) so it is neither even nor odd func

xe^x = 0 => x = 0 is root

f(x) is positive for x>0 and negative for x<0

im(x->-inifinity) f(x) = infinite

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

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

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