f(x) vs. f ' (x)

1. Function Increasing (-2,-1)U(0,1) here the slope is increasing; Function Decreasing (-1,0)U(1,2) because the slope is decreasing . Also because f '(x) is the derivative of f(x) and the derivative is the slope of the original function.
2. Local Extrema local maximum x=+1 and x=-1 there the slope was positive and then had a flat slope to turn negative. local minimum x=0 there the slope was negative then reach to a slope of 0 and then switch to a positive slope making a critical point.
3. Concave up between (-1,1) where it looks like it can hold water. Concave down is (-2,-1)U(1,2) where it looks like its going to spill water.
4. x^3+x^2 is f(x) on the given slopes and concavity shows it as the two power function are combined.