Chapter 2.7, Problem 39E

To determine

To graph: The function $f\left(x\right)=x+\sqrt{\left|x\right|}$, and conclude the differentiability of the points $\left(-1,0\right)$ and $\left(0,0\right)$.

Answer to Problem 39E

The function $f\left(x\right)$ is differentiable at $\left(-1,0\right)$ but not differentiable at $\left(0,0\right)$.

Explanation of Solution

Given:

The given function is $f\left(x\right)=x+\sqrt{\left|x\right|}$.

Graph:

Use the online graphing calculator to draw the graph of the function $f\left(x\right)=x+\sqrt{\left|x\right|}$ as shown below in Figure 1.

Check the behavior zoom in the graph toward the point $\left(-1,0\right)$.

Graph:

Use the online graphing calculator to zoom toward the point $\left(-1,0\right)$ as shown below in Figure 2.

Observation:

From Figure 2, it is observed that when zoomed in the graph toward the point $\left(-1,0\right)$, the graph of the tangent line and the curve looks likes almost identical.

Hence, the given function $f\left(x\right)=x+\sqrt{\left|x\right|}$ is differentiable at $\left(-1,0\right)$.

Check the behavior zoom in the graph toward the point $\left(0,0\right)$.

Graph:

Use the online graphing calculator to zoom toward the point $\left(0,0\right)$ as shown below in Figure 3.

Observation:

From Figure 3, it is observed that when zoomed in the graph toward the point $\left(0,0\right)$, the curve has high sharpness at origin. So, there is no unique tangent line at origin.

Thus, origin is a corner point.

Hence, the given function $f\left(x\right)=x+\sqrt{\left|x\right|}$ is not differentiable at $\left(0,0\right)$.

Conclusion:

By zooming the graph toward the point $\left(-1,0\right)$, the graph of the tangent line and the curve looks likes almost identical while zooming the graph toward the point $\left(0,0\right)$,the graph has no unique tangent line.

Thus, $f\left(x\right)=x+\sqrt{\left|x\right|}$ is differentiable at $\left(-1,0\right)$ but not differentiable at $\left(0,0\right)$.