- #1
JG89
- 728
- 1
Homework Statement
Show that the function [tex] f(x) = \frac{x^8 + 9x^9 - 12x^{13}}{1 + x^3 - x^6} [/tex] has a local minimum at x = 0.
Homework Equations
The Attempt at a Solution
If f is to have a local minimum at x = 0 then we must have f'(0) = 0. Without typing up the tedious calculations, is this the correct procedure:
[tex] f(x) = (x^8 + 9x^9 - 12x^{13}) \frac{1}{1 - (x^3 +x^6)} [/tex]. Let [tex] u = x^3 + x^6 [/tex]. Then [tex] f(x) = (x^8 + 9x^9 - 12x^{13}) \frac{1}{1-u} = (x^8 + 9x^9 - 12x^{13}) (1 + u + u^2 + ...) [/tex].
I can differentiate this using the product rule (and differentiate the expression on the right hand side term by term) or I can multiply both brackets and then differentiate term by term, remembering that u is a function of x. It seems to me for that f'(x), every term will be of the form x^i for some integer i, and so f'(0) = 0??