To determine which curve represents $f(x)$, $f'(x)$, and $f''(x)$ in the given graph, we can analyze the shape and behavior of the curves:

1. The purple curve (upward parabola): This is likely to be $f(x)$ because:

   • It has the general shape of a quadratic function, which suggests it represents the original function $f(x)$.
   • The function increases rapidly as $x$ moves away from 0, typical for a parabola.

2. The black curve (with peaks and valleys): This is $f'(x)$, the first derivative of $f(x)$, because:

   • The derivative of a quadratic function is a linear function, and the black curve exhibits some turning points.
   • The function crosses the x-axis (where the slope of $f(x)$ is 0), indicating where $f(x)$ has maxima or minima.
   • The shape has peaks and valleys which correspond to the changes in slope of the original function.

3. The red curve (flat line): This is $f''(x)$, the second derivative of $f(x)$, because:

   • The second derivative of a quadratic function is constant (i.e., the rate of change of the slope is constant). The flat red line near 0 confirms this.
   • The second derivative evaluates concavity