1. The problem asks which function among the options is increasing on the interval $]a,b[$ given that $f$ is a function defined on $[a,b]$ with values in $\overline{\mathbb{R}}$ and the graph of $f$ is shown. 2. To determine if a function $g(x)$ is increasing on $]a,b[$, we check if its derivative $g'(x) > 0$ for all $x \in ]a,b[$. 3. Let's analyze each option: (a) $g(x) = [f(x)]^2$ Calculate the derivative: $$g'(x) = 2 f(x) f'(x)$$ Since $f(x)$ and $f'(x)$ can vary in sign, $g'(x)$ is not guaranteed to be positive everywhere. (b) $g(x) = x \times f(x)$ Derivative: $$g'(x) = f(x) + x f'(x)$$ Again, without more information about $f(x)$ and $f'(x)$, we cannot guarantee $g'(x) > 0$. (c) $g(x) = [f(x)]^3$ Derivative: $$g'(x) = 3 [f(x)]^2 f'(x)$$ Since $[f(x)]^2 \geq 0$, the sign of $g'(x)$ depends on $f'(x)$. If $f$ is increasing (i.e., $f'(x) > 0$), then $g'(x) > 0$. (d) $g(x) = -2x - f(x)$ Derivative: $$g'(x) = -2 - f'(x)$$ This is negative if $f'(x) > -2$, so $g$ is not necessarily increasing. 4. From the graph description, $f$ is increasing on $]a,b[$ (the curve rises from $a$ to $b$). 5. Therefore, option (c) $g(x) = [f(x)]^3$ is increasing on $]a,b[$ because its derivative is positive when $f$ is increasing. **Final answer:** (c) $[f(x)]^3$ is increasing on $]a,b[$.