what is the increasing interval of the function f(x)= x^4-2x^3-5x^2+5x-4

Answer 1

Tori,

You indicate that you are a Precalculus student, so you would probably not be learning about derivatives (f'(x)) In your course.

If you have a graphing calculator, such as a TI 84 or 84, the you can use the CALC function to find the local minimums and local maximums of functions.

If you do, you will see the the increasing interval for this function is approximately (-1.263, 0.423) ∪ (2.340, ∞).

The endpoints given are those of irrational numbers, and quite messy ones at that.

You should always try to use the rational root theorem to locate the rational roots and then simplify the function by factoring them out. Unfortunately, this polynomial has no rational roots.

There is a quartic equation formula and a cubic equation formula, but they are quite difficult to use compared to the numerical methods programmed into calculators.

Answer 2

This function is increasing where f'(x)>0. f'(x)=4x^3-6x^2-10x+5 and we need to see at what intervals it is positive. For this reason we solve for the zeroes of f'(x)=0. Using an equation calculator we find out that the zeroes are: x1≅-1.26, x2≅0.42, x3≅2.34.

now we discuss the signs of f: f<0 for x→-∞ therefore x<-1,26 is not the answer. By this clue and by looking at the oscillating graph of a cubic function with three zeroes we understand that the answer for the problem is:

(-1.26,0.42)∪(2.34,+∞).