# How do you sketch the general shape of #f(x)=-x^5+3x^3+2# using end behavior?

##### 1 Answer

See explanation...

#### Explanation:

End behaviour will help, but it's not really enough...

The end behaviour is determined purely by the term of highest degree, that is the term

Since this term is of odd degree with negative coefficient, we find that:

#lim_(x->oo) f(x) = -oo#

#lim_(x->-oo) f(x) = oo#

If

Since this term is of odd degree with positive coefficient, we have:

#lim_(x->oo) 3x^3+2 = oo#

#lim_(x->-oo) 3x^3+2 = -oo#

... precisely the opposite of

So one question we might ask is whether the term in

The answer is yes. Since there is only a constant term following, the

Next note that the constant term

Next note that

We can supplement our analysis by looking at the derivative:

#f'(x) = -5x^4+9x^2 = -5x^2(x^2-9/5)#

Hence there is a local minimum at

You can calculate a few example points too to help find that the curve looks something like this...

graph{-x^5+3x^3+2 [-10, 10, -5, 5]}