say we have a polynomial function
$\displaystyle x^4+ux^2 + v x $ (u,v some numbers)
does anyone know why adding a higher order term to this will not change what happens near zero (in terms of critical points near zero)
say we have a polynomial function
$\displaystyle x^4+ux^2 + v x $ (u,v some numbers)
does anyone know why adding a higher order term to this will not change what happens near zero (in terms of critical points near zero)
If the function had an $\displaystyle x^5$ term, the derivative of that is $\displaystyle 5x^4.$
For very small values of x, $\displaystyle 5x^4$ will be very small...$\displaystyle 5(0.1)^4=0.0005$
hence there will be only very small changes in the derivative near x=0, if a higher-power term is added to the function
(depending on the proximity of x to zero).