A function is said to be increasing in the region where the value of the function(y) increases as we increase the value of x. A function is said to be decreasing in the region where the value of the function(y) decreases as we increase the value of x. We can clearly see the blue graph, it is that of the locus of the function f(x)=y= ax^3+bx^2+cx+d . We can change the values of a, b, c and d using their respective slide bars. Point A lies on the function f(x) and a tangent is drawn at this point. We can see that an angle α is also shown between the positive x axis and this tangent. We know that tanα is positive when α<90˚ and tanα is negative when α>90˚. α is the angle between the x axis and the tangent at point A so tanα gives the slope of the tangent. We also know that the value of df(x)/dx at the point A is equal to the slope of the tangent at that point. Let us change the values of a, b, c and d and move the point A and now observe how the angle α gets changed.

Question/s to think about.
Compare the values of α and find when is the function increasing or decreasing?.
Compare the values of df(x)/dx at the point A and find when is the function decreasing or when is the function increasing?