Critical Points – Local Minimum At X=2 In Math Functions.

the graph of f'(x) touches the x-axis at x=2 but remains above the x-axis on both sides

x=2 is a critical point, but not a relative max or min, of f(x)

The fact that the graph of f'(x) touches the x-axis at x=2 means that f(x) is neither increasing nor decreasing at that point, but rather it has a critical point or stationary point. However, since the graph of f'(x) remains above the x-axis on both sides, this implies that f(x) is experiencing a minimum point at x=2, where f(x) changes from decreasing to increasing. The reason for this is that the slope of the tangent line to the graph of f(x) is zero at x=2, indicating that the derivative is zero at this point, while the fact that the graph of f'(x) is positive on both sides implies that the derivative changes from negative to positive, suggesting a transition from decreasing to increasing.

To summarize, the graph of f(x) has a local minimum at x=2, where the derivative f'(x) is equal to zero and changes signs.

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