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The First Derivative Test is a crucial mathematical process used to determine the relative extrema of a function. When studying the behavior of a function, one of the questions often asked is where the function reaches its highest or lowest points, known as relative maxima or minima. To apply the First Derivative Test, we must first locate the function's critical points, which are points where the function's derivative is either zero or undefined.

Once the critical points are found, these points are analyzed to understand how the function's slope changes. If the derivative changes from positive to negative at a critical point, the point is a relative maximum. If the derivative changes from negative to positive, the point is a relative minimum. If there is no change, the test is inconclusive.

For the given function, \( f(x) = x^3 + x + 1 \), the derivative \( f'(x) = 3x^2 + 1 \) never equals zero, indicating there are no points at which the slope of the original function changes from positive to negative or vice versa. As such, no relative maxima or minima exist for this function, signaling a consistently increasing function.

Critical points are where a function's derivative is either zero or does not exist. These points are essential in analyzing the function's graph because they can potentially indicate the location of relative extrema like peaks or troughs. Finding the critical points is often one of the first steps in studying a function's behavior.

To find the critical points, one must calculate the derivative of the function and solve for the values of the variable that make the derivative zero or identify where the derivative is undefined. In the context of our function \( f(x) = x^3 + x + 1 \), the derivative \( f'(x) = 3x^2 + 1 \) does not have any real roots, meaning it is never zero, and it is also never undefined since it is a simple polynomial. Therefore, no critical points can be identified, and thus no relative extrema exist for this function over the entire set of real numbers.

The Power Rule is one of the most fundamental rules of differentiation, which is used to find the derivative of a function with respect to a variable. This rule applies to monomials where a real number is raised to a power. The Power Rule states that if the function is \( f(x) = ax^n \), then the derivative of the function, \( f'(x) \), is \( anx^{n-1} \).

Applying the Power Rule simplifies finding derivatives significantly, particularly for polynomial functions. For instance, in the function \( f(x) = x^3 + x + 1 \), applying the Power Rule to each term individually, we get the derivative \( f'(x) = 3x^2 + 1 \) (since the derivative of a constant is zero and the derivative of \( x \) is 1). This rule proves to be invaluable when analyzing the function for critical points as part of determining the locations of relative extrema.