Fortunately, you do not need the equation for g(t) to do this problem. Just remember that critical points of f(x) (local extrema and inflection points) occur when f'(x)=0. In one of your other problems you saw that f'(x)=g(x), so the critical points of f(x) occur when g(x)=0. Looks to me like that happens at 0, 2, and 6. I will leave it to you to classify these as local max/local min/inflection point.
Sketching the graph of f(x) is easy if you remember that f(x) is just the area between the graph of g(t) and the t-axis from 0 to x, with areas under the t-axis counting as negative. The area starts with 0 at 0, is increasingly negative until x=2, decreasingly negative between 2 and about 3.5, and positive thereafter.