Given a quintic polynomial (in my case, $x^5+2x^4+16x-32$), I am supposed to get its maximum and minimum value for the interval $I=[-2;2]$ without using the derivative of the corresponing polynomial function.
I remember that for quadratic functions, there was a trick that required you to factor your polynomial and then maximize or minimize all of the factors.
But how would I do this for polynomials of higher degrees?
We have $$ x^5+2x^4+16x-32=(x+2)(x^4+16) - 64 $$ First term is always non-negative on the given interval (hence minimum is attained at $-2$), and increasing, as is easy to show without using derivatives.