Hi guys, im stuck on a problem, this is it:
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Let f(x) = - u''(x)/u'(x)
Show that if f(x) is decreasing then u'''(x) is greater than 0.
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Thanks Guys, ive been at it for a while and cant get it
If $\displaystyle f = - \frac{u''}{u'}$ then $\displaystyle f' = - \left( \frac{u' u''' - u''^2}{u'^2} \right)$. If $\displaystyle f' < 0$ then $\displaystyle - \left( \frac{u' u''' - u''^2}{u'^2} \right) < 0 $ so $\displaystyle \left( \frac{u' u''' - u''^2}{u'^2} \right) > 0$ so $\displaystyle u' u''' - u''^2 > 0$ giving $\displaystyle u' u''' > u''^2 $ so $\displaystyle u' u''' > 0 $. You need the product. It is possible to come up with a function that $\displaystyle u' > 0\;\;\text{and}\;\;\; u'''<0$.
Case in point $\displaystyle u ' = e^{-x^2}$ so $\displaystyle f = -2x\;\;\Rightarrow\;\; f' = -2 < 0$ yet $\displaystyle u''' = (4x^2-2)e^{-x^2}$ which is not always positive for all x