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Problem 60 Hard Difficulty

Where is the greatest integer function $ f(x) = [ x ] $ not differentiable? Find a formula for $ f' $ and sketch its graph.

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let's talk about this question. It seems that where is the greatest indigent function? Not the friendship. So in the best way to answer this is by drawing the graph of the greatest indigent function. This is the coordinate access. And I'm gonna just brought up the indigenous here. So this is zero one, two, three and so on. Uh this is -1 -2 -3 -4 and so on. Over here we have one, two, 34. Uh and over here we have -1 -2 and so on. So between 0-1, between 0-1, The functional value of zero From 1 to 2, The functional value is one from 2 to 3. From 2 to 3. The functional value is two and so on. From -1-0, There's zero. The open circle means the start improving values -1 From -2 to -1, the values -2 and so on. Clearly we can see that it has a jump discontinuity at each integral value at each integral value. It has a jump kind of discontinuity so it is not differentiable at all those points. So we're going to say that it is not differentiate able at all integral whites. And saying that find the formula for the f dash and sketches craft clearly the derivative whatever the function and this function is always a constant because function between uh function between 0 to 10 function between 1 to 2 as one function between 2 to 3 uh is two and so on. Uh So wherever the functional value it gets a functional value that's a constant. Because function is uh let's call it minus two for x. Between minus two to minus one, it's minus one for x. Between minus one and 00 for X between 10 and one. It's one for X, between one and two and so on. So whenever we will take the derivative the derivative of a constant is always zero. So we're gonna see that if dash X is always zero, definitely excluding the integral points because that's where the function is different. She able and if we have to draw the graph of uh Everyday Acts Equal to zero. So why equal to zero is just the X. Axis Except the integral value. So this is how it's gonna graph is gonna look like. So we'll just make a draw the plot the integral values and we're gonna draw holes over here. So that is a whole this is a whole, this is a whole, this is a whole and so on. This is how the graph of F dash X is going to look like because F dash X is always zero. Thank you