Stationary Points

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What are stationary points in calculus?

Back

Stationary points are points on a curve where the derivative (slope) is zero, indicating a potential local maximum, minimum, or inflection point.

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How do you find stationary points of a function?

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To find stationary points, take the first derivative of the function, set it equal to zero, and solve for the variable.

Front

What is the first derivative of a function?

Back

The first derivative of a function represents the rate of change of the function with respect to its variable, indicating the slope of the tangent line at any point.

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What does it mean if the first derivative is positive?

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If the first derivative is positive, the function is increasing at that point.

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What does it mean if the first derivative is negative?

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If the first derivative is negative, the function is decreasing at that point.

Front

What is the second derivative of a function?

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The second derivative is the derivative of the first derivative, indicating the curvature of the function and whether it is concave up or down.

Front

How can the second derivative help identify stationary points?

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The second derivative test can determine the nature of stationary points: if it's positive, the point is a local minimum; if negative, a local maximum; if zero, the test is inconclusive.

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The stationary points are (0, 0) and (2, -4).

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Back

The stationary point is (1/2, 1).

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What is the significance of a stationary point in a real-world context?

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Stationary points can represent optimal solutions, such as maximum profit or minimum cost in various applications.