Use the first-derivative test to determine the local extrema...

Use the first-derivative test to determine the local extrema of each function $f$. Then verify your algebraic answers with graphs from a calculator or graphing utility.
$$
f(x)=e^{x}\left(x^{2}-x-1\right)
$$

Key Concepts

Product Rule in Differentiation

When differentiating a function that is the product of two or more functions, the product rule is applied. This rule states that the derivative of a product is the derivative of the first function multiplied by the second function plus the first function multiplied by the derivative of the second function, which is crucial for correctly finding the derivative of products like the one in the problem.

Local Extrema

Local extrema refer to the local maximum and minimum values of a function. Determining these values involves examining how the function behaves in the vicinity of its critical points, specifically by using tests like the first derivative test to confirm whether the function is increasing or decreasing around these points.

Graphical Verification

Graphical verification involves using a graphing utility or calculator to plot the function and visually inspect its behavior. This technique provides a practical check of the algebraically determined local extrema by confirming where the function actually has peaks, valleys, or inflection points on its graph.

First Derivative Test

This is a method used in calculus to determine whether a critical point is a local maximum or minimum by analyzing the sign of the derivative before and after the point. When the derivative changes from positive to negative, the function has a local maximum, and when it changes from negative to positive, the function has a local minimum.

Critical Points

Critical points are points in the domain of a function where its derivative is zero or undefined. These points are the primary candidates for local extrema because they are where the function's rate of change can shift, leading to peaks or valleys in the graph.

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