Wavy Curve Method or Methods of Intervals

The Wavy Curve Method (also known as the Method of Intervals) is a mathematical technique widely applied to solve inequalities of the form f(x)g(x)> 0, f(x)g(x)< 0, f(x)g(x) 0, or f(x)g(x) 0. By using the zeroes (roots) of the functions f(x) and g(x), a graph over different intervals is sketched. The basic certainty is that f(x)g(x) can only alter signs at zeros and asymptotes (a line that approaches the given curve but does not pass it at any finite span). Certain rules of the Wavy Curve Method must be considered and followed. 

Steps Involved

The Wavy Curve Method method comprises the following steps: 

1. All the zeroes of the given function, which are located on the left-hand side of the inequality, are supposed to be marked on the number axis as points. All the discontinuity points of the given function, which are located on the left-hand side of the inequality, are supposed to be marked on the number axis as points (make sure the points of zeros and points of discontinuity are easily distinguishable).

2. The number located on the extreme right of the number axis acts as a bracket. The value of a function more significant for any real number more significant than this number is supposed to be checked.

3.  A wavy curve must be drawn to pass through all the points marked on the number axis. If the value of the function obtained by the above step is positive, a wavy curve is drawn starting from the right side to the left above the number line. If the value of the function obtained is negative, a wavy curve is drawn starting from the right side to the left below the line. The union of the estimated interval obtained from given inequality is considered the required solution of the inequality using the Wavy Curve Method.

Illustrations Involved

f(x)g(x)=(x-p)(x-q)(x-r)(x-s)(x-t) is the provided function of x.

Applying the rules of the Wavy Curve Method, the required intervals for different inequalities can be found. Points p, q, and r are the critical points. Points s and t are discontinuity points. 

The points p, q, r, s, and t should be marked on the axis. After marking the points, intervals should be marked on the axis. + and – should be put on alternate intervals. For f(x)g(x)>0, consider the intervals containing positive signs. The critical points of the function should be kept out. For f(x)g(x)< 0, consider the intervals containing negative signs. It should be noted that the function’s critical points should be kept out. For f(x)g(x) 0, the required solution set is the intervals containing positive signs and critical points. Here discontinuity points should be kept out. For f(x)g(x) 0, the required solution set is the intervals containing negative signs along with critical points. Here discontinuity points should be kept out. 

Remark

Suppose another function with power higher than 1 is given; a sign of the function at a particular point where the power is odd should be changed. Apart from this case, the marking should not be changed. 

Conclusion

In simple terms, it can be said that the method helps find the intervals where the solutions of equations lie. The Wavy Curve Method applies to nth power equations. The technique provides a quick way to plot regions only by observing the signs of the given function in that region without the need to solve other finer characteristics. Thus, the intervals method is worth using whenever a function satisfies the conditions to apply it. It is a good option in case of multiple critical points.