Find the relative extrema of each function, if they exist....

Find the relative extrema of each function, if they exist. List each extremum along with the $x$ -value at which it occurs. Then sketch a graph of the function.
$$g(x)=x^{3}+\frac{1}{2} x^{2}-2 x+5$$

Key Concepts

Second Derivative Test

The Second Derivative Test uses the value of the second derivative at a critical point to determine the concavity of the function at that point. If the second derivative is positive, the function is concave up, indicating a local minimum; if it is negative, the function is concave down, indicating a local maximum. This test provides a quick way to classify critical points when the first derivative test is inconclusive or less straightforward.

First Derivative Test

The First Derivative Test involves analyzing the sign of the first derivative before and after a critical point to classify it as a relative maximum or minimum. If the derivative changes from positive to negative, the point is a relative maximum; if it changes from negative to positive, the point is a relative minimum. This test is a foundational tool in calculus for determining the local extremum behavior of functions.

Graphing Functions

Graphing functions involves plotting the function on a coordinate system to visualize its overall behavior, including the location of relative extrema, inflection points, and intervals of increase or decrease. This process synthesizes the information obtained from differentiation and analysis of critical points, allowing for an accurate representation of the function's behavior.

Critical Points

Critical points are values in the domain of a function where its derivative is equal to zero or does not exist. These points are significant because they often correspond to changes in the behavior of the function, such as where a function achieves a local maximum or minimum.

Differentiation

Differentiation is the process of computing the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of finding relative extrema, differentiation is used to identify where the slope of the function is zero or undefined, as these points are potential candidates for local minima or maxima.

Relative Extrema

Relative extrema refer to the local maximum and minimum values of a function. These are the highest or lowest points, respectively, in a small interval around a given point. Identifying relative extrema is an essential aspect of analyzing the behavior of functions, particularly when determining how the function changes and what its graph looks like near those points.

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