Maximum and minimum absolute of a polynomial in interval
I want to find the maximum value and minimum value absolute value of this polynomial $p(x) = -4x^4+12x^3+52x^2-108x-143$ in the interval $[-1.7,4.0]$
I don't know if i am doing it right. I proceed like this:
$$p'(x) = -16x^3+36x^2+104x-108=0 $$ $$x_1=-2.1724, x_2=0.87620, x_3=3.5462$$
So i input the function in R code and i get this points:
pol<-function(x){
-4*x^4+12*x^3+52*x^2-108*x-143
}
pol(-2.1724)
pol(0.87620)
pol(3.5462)
pol(-1.7)
pol(4.0)
> pol(-2.1724)
[1] 124.9089
> pol(0.87620)
[1] -191.9933
> pol(3.5462)
[1] 30.50625
> pol(-1.7)
[1] 98.5156
> pol(4.0)
[1] 1
So, I have that the absolute minimum is $0.87620$ and the maximum is $-1.7$?
we have $$\\{98.5156,\\{x\to -1.7\\}\\}$$ the maximum and $$\\{-191.993,\\{x\to 0.8762\\}\\}$$ the minimum