Don't know where to start on two problems

[zdawnz]

Here is the problem:

Let f be the function given by f(x)=2xe^(2x)
(a) Find Lim as x -> -00 f(x) and lim as x -> 00 f(x)
(b) Find the absolute minimum value of f. Justify that your answer is an absolute minimum.
(c) What is the range of f?
(d) Consider the family of functions defined by y=bxe^(bx), where b is a nonzero constant. Show that the absolute minimum value of bxe^(bx) is the same for all nonzero values of b.

This question was taken from an ap exam, and nowhere in my book describes anything close to figuring this out, any help is appreciated.

I have no idea where to start. This worksheet is over a chapter dealing with derivatives and integrals of logs. So I'm guess i have to differentiate this? How should i go about this? should i take the natural log of both sides?

lny=ln2+lnx+2xlne

then take the derivative of this?

y'/y=0 + 1/x + 2x
y ' = y ( (1/x) + 2x)
y ' = 2xe^(2x) ( (1/x) + 2x)

This doesn't make any sense and I think I am going around in circles.



the second problem is

Evaluate: integral [[e^(1/(x+1))]/(x+1)^2]dx

 

[royhaas]

The minimum occurs when \(\displaystyle f'(x)=0, f''(x)>0)\). For the integral, use the substitution \(\displaystyle u=1/(x+1)\).