Question

Find the intervals where the graph is concave up or down. Poins of inflection are: -squareroot2, 0, squareroot of 2. Original function is f(x)= 20x^3-3x^5 and the second derivative is f(x)= 120x-60x^3

Answer 1

the second derivative has the zeros that you wrote. and it is a third degree polynomial with negative leading coefficient. the picture (in your head) is like this

Answer 2

so it is positive, then negative, then positive, then negative again. so your function will be concave up on
\[(-\infty,-\sqrt{2})\] concave down on
\[(-\sqrt{2},0)\] etc