When does the slope of the curve y = xe^(2x) equal 0?

[Johnyi]

Homework Statement

Let f(x) = xe^(2x)

A) Find the values of x for which the slope of the cruve y = f(x) is 0
B) Explain the meaning of your answer to part (a) in terms of the graph f

Homework Equations

The Attempt at a Solution

I tried setting xe^(2x) to 0 by making it 0 = xe^(2x). I don't know what to do from there on. Am i supposed to take the derivative of xe^(2x)?

 

[RoshanBBQ]

A derivative is a fancy way of saying "instantaneous slope." So if you need to find at which values of x the slope is instantaneously 0, you need to set the derivative to zero, not the function itself.

 

[Johnyi]

A derivative is a fancy way of saying "instantaneous slope." So if you need to find at which values of x the slope is instantaneously 0, you need to set the derivative to zero, not the function itself.

So i get 2xe^(2x) = 0
The answers in the back of the book say that x = 1/2 but i just can't seem to find a way to that answer

 

[RoshanBBQ]

So i get 2xe^(2x) = 0
The answers in the back of the book say that x = 1/2 but i just can't seem to find a way to that answer

Product rule.

 

[Johnyi]

Product rule.

Im sorry i just can't seem to do this right..so far i got

x(2e^(2x))+e^(2x)
=2e^(2x)x+e^(2x)

 

[RoshanBBQ]

Im sorry i just can't seem to do this right..so far i got

x(2e^(2x))+e^(2x)
=2e^(2x)x+e^(2x)

Yes, that looks right. Now set that equal to zero and solve for x. For the record, as you've written f(x), I believe the answer in the book is incorrect.
$$2xe^{2x}+e^{2x} = 0$$

Can you solve it from here? What answer do you find? I will tell you if I find the same answer.

 

[Johnyi]

I thinking that you can factor out a e^(2x).

So it would be e^(2x)(x+1).

Then subtract 1 to the other side, then divide by...I have no clue

 

[RoshanBBQ]

I thinking that you can factor out a e^(2x).

So it would be e^(2x)(x+1).

Then subtract 1 to the other side, then divide by...I have no clue

if A(x)B(x) = 0, what must be true about at least A(x) or B(x)? Think back to algebra I where you solved

$$x^2-1=(x+1)(x-1)=A(x)B(x) = 0$$
$$A(x)=x+1$$
$$B(x)=x-1$$

Another method to arriving to the same conclusion is:
What does
$$\frac{0}{A}$$
equal if A is nonzero? Is an exponential ever zero? A/A = 1 if A is nonzero too. Can you think of a single step to remove the exponential?

 

[Johnyi]

Im sorry i don't understand

 

[RoshanBBQ]

Im sorry i don't understand

Divide both sides by the exponential or you can simply say if A(x)B(x) = 0, A(X) = 0 or B(x) = 0.
$$A(x) = e^{2x}$$
$$B(x) = 2x+1$$
A(x) cannot reach zero for a finite x. B(x) can. So solve for when B(x) = 0 for the solution. Note, you end up solving for B(x) = 0 if you divide both sides by A(x) (the exponential).