   Chapter 2.2, Problem 43E

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# Graph the function f ( x ) = x + | x | . Zoom in repeatedly, first toward the point ( − 1 , 0 ) and then toward the origin. What is different about the behavior of f in the vicinity of these two points? What do you conclude about the differentiability of f ?

To determine

To Find:

The behavior of the graph of function fx=x+x12 from x=-1 to x=0.

Solution: The function is differentiable at x = -1 and is not differentiable at  x = 0. The function is continuous at both x = -1 and x = 0. The graph is very smooth near x = -1 and is increasing.. But in the vicinity of 0 to its left the function is decreasing rapidly whereas to its right the function is increasing rapidly. Also the function appear to take a sharp turn at x = 0.

Explanation

From the graph function appears to be is continuous on the interval (-1,0). And clearly it is differentiable at x = -1.

From graph, we can see that there is a sharp turn in the graph at x = 0. Therefore it is not differentiable at x = 0. We may also conclude this from observing that the slope of tangents become more and more negative if we approach 0 from left and more and more positive as we approach 0 from right. So it is not possible to draw a unique tangent to the function at 0.

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