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gokuvaibhav · Dec 30, 2006, 10:51 AM

Q1.. Let f(x) =[a+bsinx], where x belongs to (0,Pi), a belongs to (I),b is a prime number and [.] denotes greatest integer function. Find the number of points at which f(x) is not differenciable?

galactus · Dec 30, 2006, 02:22 PM

Originally Posted by gokuvaibhav

Q1.. Let $f(x) =[a+bsinx]$ , where x belongs to $(0,{\pi})$ , a belongs to (I),b is a prime number and [.] denotes greatest integer function. Find the number of points at which f(x) is not differentiable?

What does 'I' represent? Integers?

gokuvaibhav · Dec 30, 2006, 09:06 PM

Q1.. Let , where x belongs to , a belongs to (I),b is a prime number and [.] denotes greatest integer function. Find the number of points at which f(x) is not differentiable?

I=integers

galactus · Jan 2, 2007, 10:48 AM

I don't know if you still need an answer or not, but I posed this
question on another math forum(mathgoodies.com) and got the following
response, which I think is very good.

I knew it would be non-differentiable at the 'jumping' point, but this is a more detailed answer.

"The derivative of a floor function at any point should be 0 if it is
differentiable at the point.

The derivative is undefined at the point where the floor function jumps.

When a+bsin(x) = I, where I is an integer, the function is not
differentiable.

Since a is an integer, b is prime, and sin(x) is between [0,1], when

sin(x)=1/b, 2/b, ...., b/b, we get a non-differentiable point. Therefore,

we have only 2b points: $\left(x=sin^{-1}(\frac{1}{b}), \;\ x=sin^{-1}(\frac{2}{b}) <br /> <br /> \;\ ,......., \;\ sin^{-1}(\frac{b}{b})$ and

${\pi}-sin^{-1}(\frac{1}{b}), \;\ {\pi}-sin^{-1}(\frac{2}{b}) \;\ ,......., \;\ {\pi}-sin^{-<br /> <br /> 1}(\frac{b}{b})\right)$ where the floor function is

not differentiable".

In other words, suppose 3 was the prime, then there would be 2b=2(3)=6
points where not differentiable: $sin^{-1}(\frac{1}{3}), \;\ sin^{-1}(\frac{2}{3}), \;\ sin^{-1}(\frac{3}{3}), \;\ {\pi}-sin^{-1}(\frac{1}{3}), \;\ {\pi}-sin^{-1}(\frac{2}{3}), \;\ {\pi}-sin^{-1}(\frac{3}{3})$ .

Does this help? I thought this was an interesting problem. I learned something myself. I had never given any real thought to the differentiation of floor functions. I am glad you posed this problem.