Question

Find the relative minimum of y = 4⋅x3 + 12⋅x2 - 10⋅x - 56

Round your answers to the nearest hundredth.


PS I don't own a graphing calc, so I'm unsure of if I need it to do this problem

Answer 1

Set the derivative to 0 and solve for x, substitute the value of x into the second derivative and take the x value for which the second derivative > 0 and substitute that value of x into y.

Answer 2

Do you know derivatives?

Answer 3

not exactly...

Answer 4

So use a spreadsheet like excel. Plot some values, start wide, like from -1000 to 1000 just to get the idea where it crosses the x-axis. Then narrow down your solutions. Since this is an odd function, you are guaranteed to have at least one real solution. Since this is a 3rd degree polynomial, you will have 3 roots, but you will not see all of them with the excel technique if some of them are not real (i.e. imaginary/complex).

Answer 5

We are not trying to find crossings of the x-axis, just where the values of y are the lowest. The description I have above is only if you are trying to find roots. So you don't have to worry about real/imaginary roots.

Answer 6

ybarrap - that's a good suggestion about using Excel to graph.
Even better is to use OpenOffice.org Word Processor and spreadsheets - it's free to download and I use it.

Answer 7

This is an odd polynomial and so there is a shortcut you can take.

One endpoint of the function will go to infinity and the other will go to negative infinity. Look at the highest power, 4x^3. When x is very large, this term dominates the whole equation and y just looks like y = 4x^3. What is the sign of y if x is positive? What is the sign of y if x is negative. That is what happens to this function as x goes off to infinity in each direction. You can see that as x gets more and more negative, this function goes off to negative infinity, never reaching a global minimum. There are asking for a relative minimum, which I take to mean a local minimum. And that will be finite. The spreadsheet technique then will work for you.

Answer 8

I think the problem is requiring a graphing calc, because we haven't covered derivatives yet

Answer 9

Here is what OpenOffice Spreadsheet gave me. You can easily see where the minimum is occurring. You just need to drill down further to find a more exact answer:

x y
-100 -3879056
-90 -2817956
-80 -1970456
-70 -1312556
-60 -820256
-50 -469556
-40 -236456
-30 -96956
-20 -27056
-10 -2756
0 -56
10 5044
20 36544
30 118444
40 274744
50 529444
60 906544
70 1430044
80 2123944
90 3012244
100 4118944

Answer 10

And here's a graph I generated using OpenOffice