Need someone to check my work

Apr 2013
Ramallah, Palestine
\(\displaystyle g(x) = 1/5x^5 - 8/3x^3 + 16x\)

what is the y intercept?

y = 0

Is there any symmetry?

yes odd because f(-x) = -f(x) so odd symettry

On what intervals is the function increasing?
\(\displaystyle x^4 - 8x^2 + 16 = 0

(x^2 - 4)(x^2 - 4) = 0
x +- 2\)

is it decreasing?
It is never decreasing bc: when you plug in a 0 you get a 0 which is not negative, so it can't be decreasing.

Where does the function have a local maximum?

x = none because it is never decreasing

Where does the function have a local miniomum?
x = None because again it is never decreasing.

Where is the function concave up?

Take second derivative and get
\(\displaystyle g''(x) = 4x(x^2 - 4)\)

now i need to plug in two numbers a number smaller than the horizontal asymptote and one larger than the horizontal asymptote? but how do I know what they are? Like what Do i need to do at this point, cause i am kind of lost.

Where are the inflection points?

Again I need help :(
Oct 2013
Sydney Australia
I am really rusty on this stuff but

g"(x) = 4x(x-2)(x+2)

This is a cubic finishing in the 1st and 3rd quadrants.

When the function is concave up, g"(x) must be positive which happens when -2<x<0 and when x>2
When the function is concave down, g"(x) must be negative which happens when x<-2 and when 0<x<2
So if concavity changes from negative to positive at a point doesn't that mean that the point is a point of inflection?
And, since the function is never decreasing doesn't that mean that x=-2,0, and 2 must all be points of inflection?

Also, this is an odd function so it has point symmetry but it does not have an axis of symmetry.

Like I said, I haven't done this stuff for a long time so maybe I am not right.

Also, how do you put the maths type into your question?