Volume of a parallelepiped depending on $\lambda$

Question

I've got a relatively simple calculus problem here but it has an unknown variable that I am not sure how to deal with.

Find the volume of the parallelepiped depending on $\lambda$ with;

$a = [2\lambda,2,2], b=[4,1,\lambda], c=[2,2,\lambda], where\ \lambda$ is an element of >$[-3;1]$

For which $\lambda$ element of $[-3;1]$ is the volume a maximum?

Using the formula for dealing with parallelepipeds;

$v=|a.(b$x$c)|$

I obtained;

$V(\lambda)=|2(-\lambda^2-2\lambda+6)|$

Now I am not entirely sure where to go from here and what exactly the answer i'm supposed to show is.

My 2 main questions are;

1. If I factorise the above equation and find two values for $\lambda$, do I just put them back into the equation $(-\lambda^2-2\lambda+6)$ and obtain the volume? If so, which one do I put in?
2. What is the last line asking for? (volume a maximum bit)

I hope I was clear enough in my questioning.

Answer 1

From your work (with a slight correction), $V(\lambda)=|2(-\lambda^2-2\lambda+6)|$. Thus for each value of $\lambda$ you get a corresponding volume. The shape of $V$ with respect to $\lambda$ is a parabola opening downward, on the interval in question. You want to find the value of $\lambda$ that maximizes $V$, and the corresponding maximum value of $V$.