IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

waterdemon

This is a good question

if f(x) = [ n + pSinx ] , 0 < x < , n I [ integers ]

P here is prime number , if p = 19 then find the no. of points where f(x) is

not differentiable .

delhitushar

answer is 36

waterdemon

No , it is not the answer . I will put the solution by today evening.

titun

A box function, say [ x ], where [x] denotes the greatest integer less than or equal to x, is not differentiable at the integral points or in other words, [ x ] is not differentiable when x is an integer.

Therefore, f (x) = [ n + p sinx ] will not be differentiable when n + p sinx is an integer.

0 < x <

, i.e 0 < sin x

1

Since, n is an integer, n + p sinx will be an integer only when p sinx is an integer.

Since, p is a prime = 19, p sinx will be an integer when,

sin x = 1/19, 2/19, 3/19,...........................18 / 19, 19/19

There are 18 values of x lying between 0 and

/ 2, for which
sin x = 1/19, 2/19, 3/19 ......................18/ 19
At

/ 2 , sin x = 19 / 19 = 1
Again there are 18 values of x lying between

/ 2 and

, for which
sin x = 1/19, 2/19, 3/19 ......................18/ 19

So, there are in total (18 + 1 + 18) = 37 values of x for given value of n and p = 19, for which f (x) is not differentiable.

Cheers !!!!!

karthik2007

Nice solution

waterdemon

Good solution Titun , but check out my solution here it is :

f(x) = [ n + pSinx ]

f(x) = n + [ pSinx ]

So , [ pSinx ] has to be integral .

f(x) will be discontinuous of " pSinx " will take integral values.

Therefore ,

Sinx = 1 , 1/p , 2/p , 3/p , 4/p , ................ ( p-1 ) / p { (p-1) ways }

Or

Sinx = -1 , -1/p , -2/p , -3/p , -4/p , .............- (p-1) / p { (p-1) ways }

So overall there can be

(2p - 2) ways and +1 way when pSinx = p/p at /2.

so total of (2p - 1) ways .

Therefore the number of values for which f(x) will be Discontinuous is

(2p-1) ,

Value of p = 19

Therefore , The number of values will be

=[ 2(19) - 1 ]

=37 values.

plZ rate me if you like it.

Cheers !!!!!!!!!!!!!!

karthik2007

I think there is a slight mistake water demon. psinx can never be zero in my opinion, because according to the question, x lies under a strict inequality, ie, it is strictly greater than zero, and strictly less than pi. So, the extra one comes from the situation where sinx = p/p at pi/2.

Anyway, the rest of the solution is convincing and different.

waterdemon

Ok you are right , thanks for pointing my mistake.

Here take a salute for that.

Cheers !!!!!!!!!!!!!!!!

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