can anyone explain wavy curve method
Answers
Answer:
The wavy curve method (also called the method of intervals) is a strategy used to solve inequalities of the form \frac{f(x)}{g(x)} > 0
g(x)
f(x)
>0 \left(<0, \, \geq 0, \, \text{or} \, \leq 0\right).(<0,≥0,or≤0). The method uses the fact that \frac{f(x)}{g(x)}
g(x)
f(x)
can only change sign at its zeroes and vertical asymptotes, so we can use the roots of f(x)f(x) and g(x)g(x) to sketch a graph of the function over different intervals.
Answer:
The wavy curve method is basically used to solve inequality problems. It can be applied to a wide array of problems ranging from polynomial to trigonometric expressions.
The general problem can be written as [f(x)/g(x)]>=0 or [f(x)/g(x)]<= 0
I would take a small problem and explain.
Let's say we need to solve x^2-5x+6>=0
1. Locate the roots - 2,3
2. Rearrange the expression - (x-2)(x-3)>=0
3. Start off with the left most root on the x-axis i.e. 2.
4. Start allocating alternate signs to regions once you encounter a root.
Since, before 2 everything is positive, you change signs at 2 and hence between 2 & 3 it becomes negative & post 3 becomes positive again.
5. The answer to our question is hence (-inf,2] U[3,inf).
Similarly if you have a problem ( x^2-5x+6/ x^2-5x+4) >=0
1. You would identify the roots and poles of the equation which are 2,3 and 1,4 respectively.
2. Now, go about repeating what we did for the last problem with just one cautionary measure in mind that the poles have to be removed from the answer as the expression is undefined for them.
3. Now, you have 4 numbers - 1,2,3,4.
- Start off with the left most ie. 1. All values before 1 are positive.
- At 1 you change the sign and keep on doing it every time you encounter a root or pole.
- Hence, the answer to your question is (-inf,1) U[2,3]U(4,inf). Note that the sets for roots are closed and for the poles are open.
PS: Instead of x as variable you can very well have a problem with g(x) as a variable like sinx or cosx or logx.One more thing is that the sign changes alternatively only when there is a single or odd number of repeated roots for a given value.