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.

f(x) = x³ - 13x² - 9x

검증된 단계별 안내

First, find the first derivative of the function f(x) = x³ - 13x² - 9x. The first derivative, f'(x), is obtained by differentiating each term: f'(x) = 3x² - 26x - 9.

Next, locate the critical points by setting the first derivative equal to zero and solving for x: 3x² - 26x - 9 = 0. Use the quadratic formula x = (-b ± √(b² - 4ac)) / 2a, where a = 3, b = -26, and c = -9, to find the values of x.

Once the critical points are found, calculate the second derivative of the function to apply the Second Derivative Test. The second derivative, f''(x), is obtained by differentiating f'(x): f''(x) = 6x - 26.

Evaluate the second derivative at each critical point. If f''(x) > 0 at a critical point, the function has a local minimum there. If f''(x) < 0, the function has a local maximum. If f''(x) = 0, the test is inconclusive.

Summarize the results by stating which critical points correspond to local maxima or minima based on the sign of the second derivative at those points.

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주요 개념

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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 x that satisfy the condition f'(x) = 0.

Second Derivative Test

The Second Derivative Test is a method used to classify critical points as local maxima, local minima, or saddle points. It involves evaluating the second derivative of the function at the critical points. If f''(x) > 0, the point is a local minimum; if f''(x) < 0, it is a local maximum; and if f''(x) = 0, the test is inconclusive.

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 that of nearby points, while a local minimum is where it is lower. Understanding these concepts is crucial for analyzing the behavior of functions and their applications in optimization problems.

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.f(x) = x³ - 13x² - 9x