Question

(you can click on a graph to enlarge it.)
inflection points at: $x = 0$
interval where function is concave up:
interval where function is concave down:

Explanation:

Step1: Recall concavity definitions

A function \( y = f(x) \) is concave up when \( f''(x)>0 \) and concave down when \( f''(x)<0 \). The inflection point \( x = 0 \) divides the domain.

Step2: Analyze left - hand side of inflection point

For \( x<0 \), the graph of \( y = f(x) \) is concave down (the slope of the tangent line is decreasing).

Step3: Analyze right - hand side of inflection point

For \( x>0 \), the graph of \( y = f(x) \) is concave up (the slope of the tangent line is increasing).

Answer:

Interval where function is concave up: \( (0,\infty) \)
Interval where function is concave down: \( (-\infty,0) \)