Question

The function f(x)=x^3-3x^2+2x rises as x grows very large.
True or False?

Answer 1

Think about it.
As x grows very large, x^3 grows very, very large since it is the cube of x.
Ok so far?

Answer 2

yes

Answer 3

Now look at the second term.
It is -3x^2.
You need to subtract 3x^2 from x^3.
3x^2 is also large compared to x since it is based on the square of x, but the cube of x is much larger than the square of x for large positive numbers, so the net effect is that the function is still growing as x gets larger and larger.
Finally you add 2x, which just makes f(x) larger, but since it is only based on x itself, it makes a relatively smaller difference than the cube term of the square term.

Answer 4

Therefore, you can conclude that the end behavior of this function as x approaches infinity is that that f(x) also approaches infinity.

Answer 5

Ok so the answer is True because even though there is a -3x^2, that doesn't effect it as heavily as the x^3 and the x together

Answer 6

Correct.
It doesn't even affect the x^3 term alone that much.

Answer 7

Now even though you are not asked, let;'s look at the behavior of the function as x goes very negative, as x approaches negative infinity.

Answer 8

Since the cube of a negative number is negative, as x approaches negative infinity, x^3 also approaches negative infinity but even faster than plain x since you are cubing an increasingly more negative number. Once again the leading term (the first term) has more influence than the other terms and the function approaches negative infinity as x approaches negative infinity.

Answer 9

ok that makes sense. So for example "the function f(x)=x^3-3x^2+2x rises as x grows very small" would be false because the x is actually getting larger. Since the first term is a positive x^3

Answer 10

@mathstudent55

Answer 11

Thank you