Identifying Extrema

In Exercises 19–40:

b. Identify the function’s local extreme values, if any, saying where they occur.

f(r) = 3r³ + 16r

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To identify the local extrema of the function \( f(r) = 3r^3 + 16r \), we first need to find its critical points. Critical points occur where the derivative of the function is zero or undefined.

Calculate the derivative of the function \( f(r) \). The derivative \( f'(r) \) is obtained by differentiating each term: \( f'(r) = \frac{d}{dr}(3r^3) + \frac{d}{dr}(16r) \). This results in \( f'(r) = 9r^2 + 16 \).

Set the derivative \( f'(r) = 9r^2 + 16 \) equal to zero to find the critical points: \( 9r^2 + 16 = 0 \). Solve this equation for \( r \).

Once the critical points are found, determine the nature of these points (whether they are local minima, maxima, or neither) by using the second derivative test. Calculate the second derivative \( f''(r) \) by differentiating \( f'(r) \): \( f''(r) = \frac{d}{dr}(9r^2) = 18r \).

Evaluate \( f''(r) \) at each critical point. If \( f''(r) > 0 \), the function has a local minimum at that point. If \( f''(r) < 0 \), the function has a local maximum. If \( f''(r) = 0 \), the test is inconclusive.

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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. To find them, compute the derivative of the function and solve for values of the variable where the derivative equals zero or does not exist.

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04:50

Critical Points

First Derivative Test

The First Derivative Test helps determine whether a critical point is a local maximum or minimum. By analyzing the sign of the derivative before and after the critical point, one can infer the behavior of the function: a change from positive to negative indicates a local maximum, while a change from negative to positive suggests a local minimum.

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07:09

The First Derivative Test: Finding Local Extrema

Second Derivative Test

The Second Derivative Test provides another method to classify critical points. If the second derivative at a critical point is positive, the function has a local minimum there; if negative, a local maximum. If the second derivative is zero, the test is inconclusive, and further analysis is needed.

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