Finding Critical Points

In Exercises 41–50, determine all critical points and all domain endpoints for each function.

y = x − 3x²ᐟ³

검증된 단계별 안내

Step 1: Understand the concept of critical points. Critical points occur where the derivative of a function is zero or undefined. These points are potential locations for local maxima, minima, or points of inflection.

Step 2: Find the derivative of the function y = x - 3x^(2/3). Use the power rule and the constant multiple rule to differentiate each term separately.

Step 3: Set the derivative equal to zero to find where the slope of the tangent is horizontal. Solve the equation for x to find potential critical points.

Step 4: Determine where the derivative is undefined. This occurs when the denominator of any fraction in the derivative is zero or when the derivative involves a root that results in an undefined expression.

Step 5: Identify the domain of the original function y = x - 3x^(2/3). Consider any restrictions on x that might affect the existence of critical points, such as values that make the function undefined.

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주요 개념

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for identifying local maxima, minima, or points of inflection. To find critical points, compute the derivative of the function and solve for values of x where the derivative equals zero or does not exist.

Derivative

The derivative of a function represents the rate of change of the function with respect to its variable. It is a fundamental tool in calculus used to find slopes of tangent lines, velocities, and optimize functions. For the function y = x − 3x²ᐟ³, the derivative helps identify critical points by setting it to zero or finding where it is undefined.

Domain Endpoints

Domain endpoints are the boundary values of the domain of a function, where the function is defined. These endpoints are crucial when analyzing the behavior of a function over its entire domain, especially when determining absolute extrema. For y = x − 3x²ᐟ³, consider the domain of x and identify any endpoints that might affect the function's critical points.

Finding Critical PointsIn Exercises 41–50, determine all critical points and all domain endpoints for each function.y = x − 3x²ᐟ³