Question

Graph the function \(f\left( x \right) = x + \sqrt {\left| x \right|} \). Zoom in repeatedly, first toward the point \(\left( { - 1,0} \right)\) and then toward the origin. What is different about the behaviour of \(f\) in the vicinity of these two points? What do you conclude about the differentiability of \(f\)?

Short answer

The graph of the function is:

The function is differentiable is\(x = - 1\), and not differentiable at \(x = 0\).

Step-by-step solution

Differentiability of the Function

The differentiability of any function can be defined as the nature of the function that indicates that its derivative is having any real values for all points lying within the domain of that function.

Sketch the graph and check differentiability.

To sketch the graph of the function\(f\left( x \right) = x + \sqrt {\left| x \right|} \)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(f\left( x \right) = x + \sqrt {\left| x \right|} \)in the\({Y_1}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the function\(f\left( x \right) = x + \sqrt {\left| x \right|} \)is shown below:

Thus, the graph of a given function \(f\left( x \right) = x + \sqrt {\left| x \right|} \) will be:

Clearly, at\(x = - 1\), the function is continuousin nature, whereas at \(x = 0\), the function is having a kink, so not differentiable.

Hence, the function\(f\) is differentiable at \(x = - 1\) and not differentiable\(x = 0\).