Calculus..Need Help Soon
posted by Leanna on .
If f(x)=1x^3 and f^1 is the inverse of f, how many solutions does the equation f(x)=f^1(x) have?
a)none b)one c)three d)five e)six

Work out the inverse of f(x), which should be:
f^{1}(x) = (1x)^{(1/3)}
Equate
f(x)=f^{1}(x) to get
(1x³)=(1x)^{(1/3)}
Take cube on both sides:
(1x³)³ = (1x)
Expand and cancel "1" on each side to give
x3x³+3x^{6}x^{9}=0 ...(1)
This is where it gets a little fussy.
Examine the signs of the coefficients equation and apply des Cartes rule of signs to get a minimum of three positive roots corresponding the three changes of sign.
In addition, we know that zero is a root which is nonpositive. So the total number of roots (by the des Cartes rule) is 3+2k, where k is at least 1 (because zero is a root), meaning that the number of roots is either 5, 7 or 9. Of these, only 5 is a given choice.
I hope someone else could give a more vigorous solution.
(The solution to equation one actually yields 5 real and 4 complex zeroes). 
Typo : "rigorous" instead of vigorous.