Concavity of the function $|f|$ , while $f$ is concave

Question

Prove or disprove, "if $f$ is concave function, then $|f|$ is also concave".

I know the result is false for conves function, but for concave function, I guess it is true. But I am unable to do the proof.

I have tried to show that the set $A=\{(x,y)\in \Bbb R^2 :|f(x)|\ge y\}$ is a convex set. For this, take two points $(x_1,y_1), (x_2,y_2) \in A$ and take $\lambda \in (0,1)$.

Now, $|f(\lambda x_1+(1-\lambda)x_2)|\ge f(\lambda x_1+(1-\lambda)x_2)\ge \lambda f(x_1)+(1-\lambda)f(x_2)$, as $f$ is concave. From this how to proceed? Can I get any help please?

Answer 1

Counterexample: $f(x)=x$ is concave, but $|f(x)|=|x|$ is convex (and it is not concave).

Answer 2

Counterexample:

Note that $$ f(x) = - x^2, \ \ \ (x \in \mathbf{R}) $$ is a concave function, but $$ | f(x) | = x^2, \ \ \ (x \in \mathbf{R}) $$ is a convex function.