Deduce

burgerandcheese

Junior Member
Joined
Jul 2, 2018
Messages
54
12131

Hi. I need help with part (a) only because then I will be able to proceed to part (b)
I have no idea how to "deduce that xe-ax < M/x if x > c"

This is what I've done thus far:

f(x) = x2e-ax
f '(x) = xe-ax(2 - xa)
f ''(x) = e-ax(2 - 4ax + a2x2)

They are correct because I checked my answer using an online derivative calculator.

So setting f '(x) = 0 gives me x = 0 or x = 2/a
f ''(0) = 2 > 0 so f(x) is at its minimum when x = 0
f ''(2/a) = -2/e2 < 0 so f(x) is at its maximum when x = 2/a

So now c = 2/a > 0
M = f(2/a) = 4/(a2e2) > 0

How do I continue?
 

HallsofIvy

Elite Member
Joined
Jan 27, 2012
Messages
4,946
Yes, \(\displaystyle f(x)= x^2e^{-ax}\) takes on its maximum value at c= 2/a and that maximum value is \(\displaystyle \frac{4}{a^2e^2}\). So for x> c, in fact for any value of x other than c, \(\displaystyle x^2e^{-ax}< \frac{4}{a^2e^2}= M\). Divide both sides of that inequality by x (which is positive).
 

burgerandcheese

Junior Member
Joined
Jul 2, 2018
Messages
54
Yes, \(\displaystyle f(x)= x^2e^{-ax}\) takes on its maximum value at c= 2/a and that maximum value is \(\displaystyle \frac{4}{a^2e^2}\). So for x> c, in fact for any value of x other than c, \(\displaystyle x^2e^{-ax}< \frac{4}{a^2e^2}= M\). Divide both sides of that inequality by x (which is positive).
Oh right, since M is the maximum value the inequality holds for any value other than c. Thanks
 
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