Finding the global minimum of f(x) from its derivative

Hi Tirth, let's solve this problem together. We are looking for the minimum value of the function f(x), which corresponds to the y-coordinate of the parabola's vertex.

Let's first establish a coordinate system. Based on the grid, we can identify the vertex of the parabola f(x) and use it as our reference point (x_v, y_v).

The derivative f'(x) is a linear function. By observing the dots on the line, we can determine its slope. It rises two units for every three units it moves to the right, so the slope a is two-thirds.

Integrating the derivative, we get the expression for f(x). Since f'(x) = 2/3(x - x_v), the function is f(x) = 1/3(x - x_v)^2 + y_v.

Now, we identify the intersection point of f(x) and f'(x) from the grid. Looking at the graph, this intersection occurs at a point where x - x_v equals 3.

At this intersection point, the y-values are equal. Let's substitute x - x_v = 3 into both equations.