This is an exercise I saw in my last test, I've been practicing it but I'm currently a bit lost when finding the roots, here's the exercise:
- Given $f:f(x)=-x^2+2x+3$, be F/F(x) is a primitive of f such as F(3)=0; analyze and graph the function h:h(x)=Ln|F(x)|.
What I did so far was integrate f(x), obtaining $F(x)=\frac{(-x^3)}{3}+x^2+3x-9$ , 9 being the constant that makes F(3)=0 possible.
From that, I proceeded with the study of $h(x)=Ln|\frac{(-x^3)}{3}+x^2+3x-9|$
Given its absolute value, after setting h(x)=0 to find the roots that pass through the x axis, I find myself with 2 equations:
$i)\frac{(-x^3)}{3}+x^2+3x-9=1$ and $ii)\frac{(-x^3)}{3}+x^2+3x-9=-1$ which, after being rearranged, end up looking like this:
$i)-x^3+3x^2+9x-30=0$ and $ii)-x^3+3x^2+9x-24=0$
So, the problem now for me would be how to find the roots of these cubic equations, the first thing that comes to mind would be Newton-Raphson to approximate the roots, but I haven't had much experience using this method which is the primary reason I came here, not for an explanation of how to use it since I already know, but more about, HOW to efficiently know where to look for the roots, I thought about just using the calculator to find the roots through the EQN mode and from there, using the integer closest to the left of the root shown in the calculator as the first $x_n$ for Newton-Raphson. Am I missing a different method? Since this is too dependent on the calculator and I'm not too used yet to "approximation" given I've always been used to fractions instead of cutting it down to a few decimals. Edit: Forgot to mention I wasn't able to find evident roots, so I backed down on that and went back to Newton-Raphson.
That said, any help is appreciated!