What characteristics of the graph of a function can we discuss by using the concept of differentiation (first and second derivatives).
(1)Using the first derivative we can tell the intervals where a function is increasing or decreasing -- a positive 1st derivative indicates the function is increasing while a negative 1st derivative indicates the function is decreasing.
We can also find extrema (maxima, minima) -- extrema only occur when the 1st derivative is zero or fails to exist. Locating these points and applying the 1st derivative test tells us whether we have a local maximum or local minimum. By using our knowledge of the intervals where the function is increasing/decreasing, we can also say if the extrema are global (absolute maximum/minimum).
(2) Using the 2nd derivative we can discuss concavity -- how the function is curved. (This is the rate of increase/decrease.) If the function is increasing and concave up, it is increasing rapidly (like an exponential) but if it is increasing but concave down the rate of growth is slowing (like a square root function)
We can also find points of inflection; this is where the graph of a function changes concavity.
The second derivative test can be used to identify whether an extrema indicated by the first derivative is a maxima,minima, or point of inflection. (If the graph is concave down the point is a maximum, if concave up a minimum.)
Let the function whose graph is drawn be y = f(x).
The first derivative of a fuction f'(x) gives us the slope of a function, and hence its graph. if it is equal to zero at a certain point then it tells that the slope of the function at that point has become parallel to the x-axis.
The second derivative f"(x) at that point tells us about the rate of change of the slope at that point.
If f"(x) is negative where f'(x) = 0, then it exhibits the maxima condition on the graph, i.e. graph makes a peak at that point
If f"(x) is positive where f'(x) = 0, then it exhibits the minima condition on the graph, i.e. graph makes a valley at that point
If f"(x) is also zero where f'(x) = 0, then it exhibits a point of contra-flexture, i.e. the graph gets parallel to the x-axis but resumes its upwards or downwards direction beyond this point.