Question

What characteristics of the graph of a function can we discuss by using the concept of differentiation (first and second derivatives).

Answer 1

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.

Answer 2

The first derivative of a graphed function is often thought of as the slope of a tangent line. The second derivative is often thought of as the convexity of the curve.