Given+the+graph+of+f+′(x),+sketch+a+possible+graph+of+f(x)

In order to sketch a possible graph of f(x) from the graph on f'(x), it is important to understand what the graph on f'(x) tells us.

1) Anywhere the graph of f'(x) hits the x axis, it means there is a horizontal tangent on the graph of f(x) at this point 2) If the graph of f'(x) passes through the x-axis from positive to negative, it means there is a relative maximum on the graph of f(x) 3) If the graph of f'(x) passes through the x-axis from negative to positive, it means there is a relative minimum on the graph of f(x) 4) Where f'(x) is positive, the graph of f(x) is increasing 5) Where f'(x) is negative, the graph of f(x) is decreasing 6) Where the graph of f'(x) has a relative minimum or maximum, the graph of f(x) crosses the x axis.

Knowing all of this, making a sketch of f(x) is simple. Where the graph of f(x) is increasing, it has a positive slope, and where it is decreasing, it has a negative slope. The maximums and minimums can be found using the rules, as well as the points where f(x) crosses the x-axis.