The vertex of the graph of f(x) = x^3 – 6 is located at?

The function f(x) = x³ – 6 does not have a vertex in the traditional sense, as it is a cubic function. Unlike quadratic functions, which are parabolas and have a single vertex, cubic functions can have various shapes, including inflection points.

If you’re looking for critical points or points of interest, we can analyze the function a bit further. For cubic functions like this one, we often find local extrema by looking at the first derivative. Let’s do that:

• First, we find the first derivative: f'(x) = 3x².
• Setting the first derivative to zero gives us 3x² = 0, which implies x = 0.

This indicates a point where the slope of the function is zero, but it is actually an inflection point rather than a local maximum or minimum. To find the function value at this point:

• f(0) = 0³ – 6 = -6.

Thus, while we cannot talk about a traditional vertex, the critical point of f(x) = x³ – 6 occurs at (0, -6), which is the inflection point of the graph.