Calculus/Calculus

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Expert: - 2/27/2008


Question
Let f be the function defined by f(x)= 2xe^-x for all real numbers of x.

A.) Write an equation of the horizontal asymptote for the graph of f.
B.) Find the x-coordinate of each critical point of f.  For each such x, determine whether f(x) is a relative maximum, a relative minimum, or neither.
C.) For what values of x is the graph of f concave down?
D.) Using the results found in parts a, b, and c, sketch the graph of y= f(x)

Thank you.

Answer
A) The horizontal assymptote can be found by noting that f(x) is always positive.  For large x, the function gets close to 0.

B) Take the derivative and set it equal to 0.  Note the graph is defined for all x.

C) When the second derivative is negative, the function is concave down.  That is why a negative second derivative at a 0 derivative is a maximum.

D) Go for it!  Hopefully you've got it.

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