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) = 2x² ln x - 11x²

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First, find the first derivative of the function \( f(x) = 2x^2 \ln x - 11x^2 \). Use the product rule for the term \( 2x^2 \ln x \) and the power rule for \( -11x^2 \).

Set the first derivative equal to zero to find the critical points. Solve the equation for \( x \) to identify potential critical points.

Next, find the second derivative of the function \( f(x) \). Differentiate the first derivative to obtain the second derivative.

Evaluate the second derivative at each critical point found in step 2. This will help determine the concavity at those points.

Apply the Second Derivative Test: If the second derivative at a critical point is positive, the function has a local minimum there. If it is negative, the function has a local maximum. If the second derivative is zero, the test is inconclusive.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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. 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. If the second derivative at a critical point is positive, the function has a local minimum there; if negative, it has a local maximum. If the second derivative is zero, the test is inconclusive, and further analysis is needed.

Natural Logarithm Function

The natural logarithm function, denoted as ln(x), is the logarithm to the base e (approximately 2.718). It is defined for positive x values and plays a crucial role in calculus, particularly in functions involving growth and decay. Understanding its properties is vital when differentiating functions that include ln(x), as it affects the behavior of the function and its derivatives.

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) = 2x² ln x - 11x²