# What are the local extrema of f(x)= (3x^3-2x^2-2x+43)/(x-1)^2+x^2?

Jan 14, 2017

Minima f : 38.827075 at x = 4.1463151 and another for a negative x. I would visit here soon, with the other minimum..

#### Explanation:

In effect, f(x )= (a biquadratic in x)/${\left(x - 1\right)}^{2}$.

Using the method of partial fractions,

$f \left(x\right) = {x}^{2} + 3 x + 4 + \frac{3}{x - 1} + \frac{42}{x - 1} ^ 2$

This form reveals an asymptotic parabola $y = {x}^{2} + 3 x + 4$ and a vertical asymptote x = 1.

As $x \to \pm \infty , f \to \infty$.

The first graph reveals the parabolic asymptote that lies low.

The second reveals the graph on the left of the vertical asymptote, x

= 1, and the third is for the right side. These are befittingly scaled to

reveal local minima f = 6 and 35, nearly Using a numerical iterative

method with starter ${x}_{0}$=3, the ${Q}_{1}$ minimum f is 38.827075 at

x =4.1473151, nearly. I would get soon, the ${Q}_{2}$ minimum.

graph{(x^2+3x+4+3/(x-1)+42/(x-1)^2-y)(x+.0000001y-1)(y-x^2-3x-4)=0 [-10, 10, 0, 50]}

graph{(x^2+3x+4+3/(x-1)+42/(x-1)^2-y)(x+.0000001y-1)=0 [-10, 10, -10, 10]}

graph{(x^2+3x+4+3/(x-1)+42/(x-1)^2-y)(x+.0000001y-1)=0 [0, 10, 0, 50]}