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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
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All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.3.89
Chapter 4, Problem 4.3.89

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


p(t) = 2t³ + 3t² - 36t

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First, find the first derivative of the function \( p(t) = 2t^3 + 3t^2 - 36t \). This involves using the power rule for differentiation. The first derivative \( p'(t) \) is calculated as follows: \( p'(t) = \frac{d}{dt}(2t^3) + \frac{d}{dt}(3t^2) - \frac{d}{dt}(36t) \).
Simplify the expression for the first derivative: \( p'(t) = 6t^2 + 6t - 36 \).
To find the critical points, set the first derivative equal to zero and solve for \( t \): \( 6t^2 + 6t - 36 = 0 \). This is a quadratic equation, which can be solved using the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 6 \), \( b = 6 \), and \( c = -36 \).
Next, find the second derivative of the function \( p(t) \) to apply the Second Derivative Test. Differentiate \( p'(t) = 6t^2 + 6t - 36 \) to get \( p''(t) = 12t + 6 \).
Evaluate the second derivative at each critical point found in step 3. If \( p''(t) > 0 \), the function has a local minimum at that point. If \( p''(t) < 0 \), the function has a local maximum. If \( p''(t) = 0 \), the test is inconclusive.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical Points

Critical points of a function occur where its first derivative is zero or undefined. These points are essential for identifying potential local maxima and minima, as they represent locations where the function's slope changes. To find critical points, one must differentiate the function and solve for the values of the variable that satisfy the condition of the first derivative being zero.
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Critical Points

Second Derivative Test

The Second Derivative Test is a method used to classify critical points as local maxima, local minima, or saddle points. By evaluating the second derivative of the function at a critical point, if the result is positive, the point is a local minimum; if negative, it is a local maximum; and if zero, the test is inconclusive. This test provides a more refined analysis of the function's behavior around critical points.
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The Second Derivative Test: Finding Local Extrema

Local Maxima and Minima

Local maxima and minima refer to the highest and lowest points in a specific neighborhood of a function's graph. A local maximum is a point where the function value is greater than the values of the function at nearby points, while a local minimum is where the function value is lower. Understanding these concepts is crucial for analyzing the overall shape and behavior of the function, particularly in optimization problems.
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The First Derivative Test: Finding Local Extrema
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