Differentiate f(x) = x^3 e^x, find intervals of increase, decrease....

[JSmith]

Given the function

.
a) Determine the intervals of increase and decrease.
b) Determine the absolute minimum value of f(x).

Can someone show me how to differentiate a function with e in it? I know its derivative is itself, but if someone could walk me through the steps it would be great!

 

[Deleted member 4993]

Given the function

.
a) Determine the intervals of increase and decrease.
b) Determine the absolute minimum value of f(x).

Can someone show me how to differentiate a function with e in it? I know its derivative is itself, but if someone could walk me through the steps it would be great!

Let me do a similar problem


\(\displaystyle f(x) \ = \ x^m * e^{n*x^p}\)

\(\displaystyle f'(x) \ = \ [(m \ * \ x^{m-1}) \ * \ e^{n \ * \ x^p}] \ + \ [x^m \ * \ (e^{n*x^p}) \ * \ (n \ * \ p \ * x^{p-1})]\)

 

[JSmith]

Let me do a similar problem


\(\displaystyle f(x) \ = \ x^m * e^{n*x^p}\)

\(\displaystyle f'(x) \ = \ [(m \ * \ x^{m-1}) \ * \ e^{n \ * \ x^p}] \ + \ [x^m \ * \ (e^{n*x^p}) \ * \ (n \ * \ p \ * x^{p-1})]\)

Great, that's what I had. Next, when I determine when the derivative is greater/less than 0, I am having some trouble. This is simple when there is only one non-numeric value, but how do I do this with e?

I have: [FONT=Trebuchet MS, Verdana, Geneva, Arial, Helvetica, sans-serif]x³e^x+3x²e^x > 0[/FONT]

 

[DrPhil]

Great, that's what I had. Next, when I determine when the derivative is greater/less than 0, I am having some trouble. This is simple when there is only one non-numeric value, but how do I do this with e?

I have: x³e^x+3x²e^x > 0

The derivative is correct, BUT why do you say ">0"? Factor the expression:

......\(\displaystyle \displaystyle f'(x) = x^2\ (x + 3)\ \mathrm e^x \)

If any factor in the numerator is zero, the expression is zero.

 

[Deleted member 4993]

Remember that e^x > 0 for all real x.

 

[lookagain]

The derivative is correct, BUT why do you say ">0"? Factor the expression:

......\(\displaystyle \displaystyle f'(x) = x^2\ (x + 3)\ \mathrm e^x \)

If any factor in the > > numerator is zero < < , the expression is zero.

DrPhil,

to what fraction are you referring by typing "numerator?" (I ask this because I don't
see any fraction in the question nor in the steps to solving it.)

 

[DrPhil]

DrPhil,

to what fraction are you referring by typing "numerator?" (I ask this because I don't
see any fraction in the question nor in the steps to solving it.)

Just an attempt to be more general, for the student to recognize in cases of rational expressions as well as this specific question.