Finding stationary points

Apr 2010
487
9
The question:
Find and identify the stationary points for \(\displaystyle y = e^{2x}(1 - x)\)

My attempt:
I differentiated the function as per here.

But I'm not sure how to solve for x when \(\displaystyle y^{'} = e^{2x}(3 - 2x) = 0\). I tried using Wolfram Alpha, but it said there are no solutions. My text states that there's a minimum point at (1/2, e/2), however.

How can I solve this? Assistance is appreciated.
 
Sep 2008
1,261
539
West Malaysia
The question:
Find and identify the stationary points for \(\displaystyle y = e^{2x}(1 - x)\)

My attempt:
I differentiated the function as per here.

But I'm not sure how to solve for x when \(\displaystyle y^{'} = e^{2x}(3 - 2x) = 0\). I tried using Wolfram Alpha, but it said there are no solutions. My text states that there's a minimum point at (1/2, e/2), however.

How can I solve this? Assistance is appreciated.
dy/dx=e^(2x)(-1)+(1-x)(2)(e^(2x))=e^(2x)[1-2x]

So when you set that to be 0, either e^(2x)=0 or 1-2x=0

but e^(2x) cannot be 0, so the only solution is 1/2.
 
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Ackbeet

MHF Hall of Honor
Jun 2010
6,318
2,433
CT, USA
With your corrected derivative, and using this command, I was able to get WolframAlpha to produce the x value that solves the equation (although, really, it's quite straight-forward to solve by inspection: the exponential is never zero, and the linear multiplier term (1-2x) is zero when x = 1/2. Done.) What command did you use to try to find the root?

[EDIT] This is the same as mathaddict's post, essentially.
 
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