Given the graph of f'(x) with f(x) continuous on all reals, find the intervals f(x) is concave up and down and find the x values of the points of inflection.

Answer 1

concave up means f" > 0
That means f' is increasing.

inflection points are where f"=0 and f'≠0

Answer 2

You need to make sure that there are actually inflection points by plugging in values near the x solution into the function of x. Inflection points may or may not be present despite finding x solutions or undefined "points" when the second derivative is set to zero.