How do you sketch a graph of f(x) and f " (x) when you have f ' (x) graph but not an equation.?

Answer 1

You have to think about it... ;-)

To graph f(x), you know you will have maxima and minima when the graph of f'(x) passes through 0, points of inflection when f'(x) has a maximum or minimum, etc. This will give you the general shape of the graph, remembering of course that in between the value of f'(x) gives you the slope at the given point. Naturally f(x) can be arbitrarily shifted up or down without changing anything (i.e. you can add an arbitrary constant).

To graph f"(x), it's pretty much the reverse process. Maxima of f'(x) correspond to f"(x) changing from positive to negative, minima of f'(x) correspond to f"(x) going from negative to positive. Inflection points of f'(x) correspond to maxima or minima of f"(x). In general, the slope of f'(x) at any point is the value of f"(x) at that point.