number of real solution of the equation sin(6pix)=x are? please clearly describe the method

Hi Rupali,

sin(Πx) = x

Now consider the graphy of

y = x, and

y = sin(Πx)

They will clearly intersect at three points.

To visualise this, consider sin(Πx) for x=1/2, it is 1.

But y=x, is 1/2 for x=1/2. So y=x is less that y = sin(Πx)

So y=x, will intersect the graph of y=sin(Πx) to the right of x=1/2.

And also similar thing will happen at the left of y-axis.

Both the graphs also pass through the origin.

So, they intersect in 3 points, and hence 3 solutions.

Hope that helps.

Best Regards,

Ashwin (IIT Madras).

Hi Rupali,

The above was the method for the number of solutions of

sin(Πx) = x.

In case of sin(6Πx) = x,

Draw the graph, you will get three intersection points to the right of y-axis ------ (because x=1/12, 5/12, 9/12 are all less than 1, and at those points sin(6Πx) will be = 1).

Similarly, there will be three intersection points to the left of y-axis.

And one at the origin.

Hence there are 7 solutions for this equation.

Hope it helps.

Best Regards,

Ashwin (IIT Madras).

There exist 11 solution of the equation as the y=sin(6pix) curve is intersected by the  y=x at 11 places. 

Hi Rupali,

Sorry for the mistake in counting,

As you can see from the above graph, it is 11 points of intersection.

5 to the right of y-axis:

1 for x= 1/12

2 for x= 5/12

and 2 for x=9/12

Similarly to the left of y-axis (5 points)

And one at the origin.

So total of 11 points of intersection.

Best Regards,

Ashwin (IIT Madras).