Question

let f(x) = -x^4 - 3x^3 + 3x + 7. find the open intervals on which f is concave up, and concave down. then determine the x-coordinates of all inflection points of f.

Answer 1

Take second derivative, find where f" is positive and where it is negative. It is concave up in the parts of the domain where it is positive, and concave down in those parts of the domain where it is negative. Points of inflection happen when f" is zero.

Answer 2

find the first derivative, and equate it to zero

-4x^3 - 9x^2 + 3 = 0

find the values of x
http://www.wolframalpha.com/input/?i=-4x%5E3+-+9x%5E2+%2B+3+%3D+0

now these points are where you gonna have concave up or concave down, take second derivative put the values on x on this,

if negative, then you will have concave down,
if positive, then you will have concave up

Answer 3

My last sentence might be bit too general, but all the points of inflection will happen at an x value where f" =0.