Question

x^3-x^2-4x+4

As x approaches infinity... y approaches..?

As x approaches negative infinity, y approaches..?

How do you determine this WITHOUT "googling" the graph?

Also, what is the maximum number of local extrema (maxima or minima) the graph of the function can have?

Answer 1

just think about it you substitute 1 for x you get y = 0

x = 10 you get y = 10^3 - 10^2 - 4x10 + 4 = 864

x = 100 you get y = 100^3 - 100^2 - 4x100 + 4 you can work this one out...

so based on these 3 values what is happening to y?

looking at negative infinity

see what happens when x = -1, -10 and =-100

same process what is happening to y? as x moves towards negative infinity...

Answer 2

Thank you! If the x value is increasing, then the y value is also increasing.
if x approaches negative infinity, then the y value would also approach negative infinity.. But are there any restrictions?

Answer 3

On the negative side of the graph, the max seems to be 6... So should I say something like
[6, infinity)?

Answer 4

this is getting pretty confusing. also, how could we determine how many local extrema the graph can have just by looking at the equation?

Answer 5

What's the degree of the given polynomial ?

Answer 6

@ganeshie8 3

Answer 7

Degree of 3 tells us the curve can cut the x axis at most 3 times

Answer 8

Next, you must ask "why" the degree has anything to do with the number of times the curve cuts the x axis.

Answer 9

Save that for later.
For now, can you figure out the number of extrema (peaks and valleys) given that the curve cuts the x axis exactly 3 times ?

Answer 10

Would that be 3-1=2?

Answer 11

Yes. Also, isn't it clear form the graph ?