Question

If $f(x)=\min \left\{1,x^2,x^3\right\}$, then

• A. $f(x)$ is continuous everywhere.
• B. $f(x)$ is continuous and differentiable everywhere.
• C. $f(x)$ is not differentiable at two points.
• D. $f(x)$ is not differentiable at one point.

Answer 1

Answer: (A), (D)

We have
$f(x)=\min \left\{1,x^2,x^3\right\}$
Let us draw the graph, which is as shown in the following figure:

The graph of minimum is shown as follows:

Now, it is obvious from the graphs that $f(x)$ is continuous throughout and non-differentiable at the point $x=1$.