find the intervals on which f is increasing and decreasing
f(x)=(x-2)/(X^2-x+1)^2
I found the derivative a got a whole mess of things:
((x^2-x+1)^2-2(x^2-x+1)(2x-1)(x-2))/(x^2-x+1)^4
Do I have to use the quadratic formula to solve for the vertices to find where it is increasing and decreasing now?
You need to simplify your derivative a bit. It boils down to
-3(x^2-3x+1)/(x^2-x+1)^3
That's a bit easier to work with. See where that takes you.
I cannot figure out how to simplify this.
okay
(x^2-x+1)-2[(x^2-x+1)(2x-1)(x-2)]/(x^2-x+1)^4
(x^2-x+1)-2[(2x-1)(x-2)](x^2-x+1)/(x^2-x+1)^4
factor out (x^2-x+1)
(x^2-x+1)[-2(x-1)(x-2)]/(x^2-x+1)^4
-2(x^2-3x+2)/(x^2-x+1)^3
plz do wait for mathematician steve to come complete it for you
but that the simplification am sure
Once you have convinced yourself that the derivative really is
-3(x^2-3x+1)/(x^2-x+1)^3
f(x) is increasing where f'(x) is positive
Note that the denominator is always positive, so f' is positive when the numerator is positive. That is, when x^2-3x+1= is negative.
That is a parabola which opens upward, so it is negative between the roots. So, on the interval
((3-β5)/2,(3+β5)/2) or (0.38,2.62) is positive, meaning f is increasing, and decreasing elsewhere.
A look at the graph confirms this.
http://www.wolframalpha.com/input/?i=(x-2)%2F(x%5E2-x%2B1)%5E2
Note the local extrema, and f is increasing between the min and the max.
To find the intervals on which the function f(x) = (x-2)/(x^2-x+1)^2 is increasing or decreasing, you should first simplify the derivative expression you found. The derivative can be simplified as follows:
((x^2-x+1)^2 - 2(x^2-x+1)(2x-1)(x-2))/(x^2-x+1)^4
To make the calculations easier, let's denote g(x) = (x^2-x+1)^2. Now we have:
(g(x) - 2(x^2-x+1)(2x-1)(x-2))/g(x)^2
Next, we need to find the critical points by setting the derivative equal to zero:
(g(x) - 2(x^2-x+1)(2x-1)(x-2))/g(x)^2 = 0
Since the denominator g(x)^2 cannot be zero, we can set the numerator equal to zero:
g(x) - 2(x^2-x+1)(2x-1)(x-2) = 0
Now, let's simplify this equation:
g(x) = 2(x^2-x+1)(2x-1)(x-2)
You can expand the expression on the right-hand side and then simplify, which will give you a polynomial equation. You can then use various methods to solve this equation, including the quadratic formula if necessary.
Once you find the critical points by solving the polynomial equation, you can determine the intervals of increase and decrease based on the sign of the derivative in those regions. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing.
Note that finding the critical points and determining intervals of increase and decrease can be a complex process, especially with higher-order polynomials. It requires careful algebraic manipulation and analysis of the sign of the derivative.