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The first derivative of a function is a crucial tool to determine where a function might change its behavior. Specifically, it helps in identifying critical points, which are potential locations of relative extrema (maximums or minimums). The process starts by taking the derivative of the function with respect to its variable. This involves applying differentiation rules, such as the power rule, product rule, or chain rule as required.

For the function given, \[g(x) = 2x^3 - 6x + 3,\] we apply the power rule to find its first derivative:

• Derive each term: \(\frac{d}{dx}(2x^3) = 6x^2\)
• \(\frac{d}{dx}(-6x) = -6\)
• \(\frac{d}{dx}(3) = 0\)

The resulting first derivative is:\[g'(x) = 6x^2 - 6.\] Critical points are found by setting the first derivative equal to zero and solving for \(x\), or identifying where it is undefined. This gives you possible points where the function's slope is zero, indicating a possible change in direction.

Once the critical points are discovered using the first derivative, the next step is to use the second derivative to determine whether these points are relative maximums, minimums, or neither.

The second derivative is simply the derivative of the first derivative. It provides information about the concavity of the original function, which can be interpreted as:

• If \(g''(x) > 0\) at a critical point, the function is concave up, indicating a relative minimum at that point.
• If \(g''(x) < 0\), the function is concave down, indicating a relative maximum.
• If \(g''(x) = 0\), the test is inconclusive, and another method is required.

For the function \(g(x) = 2x^3 - 6x + 3\), we have:\[g'(x) = 6x^2 - 6.\] The second derivative is:\[g''(x) = 12x.\] By substituting the critical points found into the second derivative, we determine the nature of these points, revealing the function's behavior more clearly.

A relative maximum is a point where the function changes direction from increasing to decreasing, creating a local "peak." It means that, within a small neighborhood, this point has the highest value.

In the context of the function \(g(x) = 2x^3 - 6x + 3\), we found that at critical point \(x = -1\), the second derivative is:\[g''(-1) = 12(-1) = -12.\] Since the second derivative is negative, it implies the function is concave down at \(x = -1\), confirming a relative maximum.

This tells us that as the graph approaches this point, it peaks and then starts to decline. It's important to visualize this to understand how the function behaves in this region.

A relative minimum occurs when the function changes direction from decreasing to increasing, creating a local "valley." This means that, locally, it is the lowest point compared to nearby values.

For the function \(g(x) = 2x^3 - 6x + 3\), upon examining the critical point \(x = 1\), we find:\[g''(1) = 12(1) = 12.\] Because the second derivative is positive, the function is concave up at this point, indicating a relative minimum.

To better visualize, imagine the graph dips at \(x = 1\) and then rises, highlighting this minimum as a crucial point of rising trend shift in its neighborhood.