Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

k(x) = x³ + 3x² + 3x + 1, −∞ < x ≤ 0

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To find the local extrema of the function k(x) = x³ + 3x² + 3x + 1 on the domain (-∞, 0], we first need to find the critical points. This involves taking the derivative of k(x) and setting it equal to zero.

Calculate the derivative of k(x): k'(x) = 3x² + 6x + 3. This derivative will help us find the critical points where the slope of the tangent is zero or undefined.

Set the derivative equal to zero to find the critical points: 3x² + 6x + 3 = 0. Solve this quadratic equation for x to find the critical points within the given domain.

Use the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, where a = 3, b = 6, and c = 3, to solve for x. This will give you the potential critical points.

Evaluate the function k(x) at the critical points and at the endpoint x = 0 to determine the local extrema. Compare these values to identify the local maximum and minimum values within the domain (-∞, 0].

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are potential locations for local extrema (maximum or minimum values). For the function k(x) = x³ + 3x² + 3x + 1, finding the derivative and solving k'(x) = 0 will help identify critical points within the given domain.

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Critical Points

First Derivative Test

The First Derivative Test helps determine whether a critical point is a local maximum, minimum, or neither. By analyzing the sign changes of the derivative around the critical points, we can infer the behavior of the function. If the derivative changes from positive to negative, the point is a local maximum; if it changes from negative to positive, it's a local minimum.

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The First Derivative Test: Finding Local Extrema

Domain Consideration

Understanding the domain is crucial when identifying extrema, as it limits where extrema can occur. For k(x) = x³ + 3x² + 3x + 1, the domain is −∞ < x ≤ 0, meaning we only consider critical points and endpoints within this range. The endpoint x = 0 must also be evaluated to determine if it is an extremum.

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