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) = 2x³ - 3x² + 12

검증된 단계별 안내

First, find the first derivative of the function f(x) = 2x³ - 3x² + 12. The first derivative, f'(x), is obtained by differentiating each term: f'(x) = 6x² - 6x.

Next, locate the critical points by setting the first derivative equal to zero and solving for x: 6x² - 6x = 0. Factor the equation: 6x(x - 1) = 0, which gives the critical points x = 0 and x = 1.

Now, find the second derivative of the function to apply the Second Derivative Test. Differentiate the first derivative: f''(x) = d/dx(6x² - 6x) = 12x - 6.

Evaluate the second derivative at each critical point. For x = 0, calculate f''(0) = 12(0) - 6 = -6. Since f''(0) < 0, the function has a local maximum at x = 0.

For x = 1, calculate f''(1) = 12(1) - 6 = 6. Since f''(1) > 0, the function has a local minimum at x = 1.

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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 optimizing values in various applications.

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) = 2x³ - 3x² + 12