Antiderivative of a Quartic Function

[ohlhauc1]

Homework Statement

I need to find information pertaining to the antiderivative of a function. At the moment, I just need help in finding part of the derivative: what is happening at the double root (on the derivative)

I am given a quartic function labelled f' with roots at: (-5,0), a double root at (-2,0), and (2,0). There are two local maximums at (0,3) and (-3.5, 3). There is a local minimum at (-2, 0). I need to find f, which I know must be a quintic function. The picture of the quartic that I am given looks like an upside down W.

Homework Equations

I am not given the equation of the quartic function.

The Attempt at a Solution

I know that for f (the quintic function), there is a minimum at x=-5 because of the root (on the quartic) and that there is a maximum at x=2 because of the root. I also know that there are two inflection points at x=-3.5 and x=0 because the derivative (quartic) has a maximum there.

I just need to determine what is present on the antiderivative at the double root at (-2,0) which is also a local maximum. According to the principles that I know, there should be a local maximum or minimum, and that it should be an inflection point on the antiderivative. However, whenever I attempt to draw the antiderivative I can fit both of those criteria on the graph. Could you help me determine what exactly the antiderivative looks like or behaves like at the point?

 

[Gib Z]

Homework Equations

I am not given the equation of the quartic function.

Yes actually, you are. You are told all 4 roots of a polynomial degree 4.

EG $$x^2+2x+1=(x+1)(x+1)$$ Hence a double root at -1. We have the roots, work backwards. The equation is

$$f'(x)=(x+5)(x+2)^2(x-2)$$. Expand and use the power rule.