Question

Find maxima and minima value
X⁴+2X³-3X²-4X+4

Answer 1

Answer:

Let f(x)=X⁴+2X³-3X²-4X+4 . Then we know, from computing the limits at ±∞ , that f must have an absolute minimum.

The derivative is f′(x)=4x3+6x2−6x−4 , which we can factor by observing that f′(1)=4+6−6−4=0 . After performing the division, we find that f′(x)=2(x−1)(x2+5x+2) and then the full factorization is f′(x)=2(x−1)(x+2)(x+1/2)

and so the derivative vanishes at −2 , −1/2 and 1 . Since the function is decreasing over (−∞,−2] we see that it is increasing over [−2,−1/2] , decreasing over [−1/2,1] and increasing over [1,∞) .

Thus the points of minimum are −2 and 1 ; the point of maximum is −1/2 . Also

f(−2)=16−16−12+8+4=0,f(1)=1+2–3−4+4=0

If you do the substitution x=t+1/2 the polynomial becomes

g(t)=t4−92t2+8116

which is much easier to work with.