Finding Critical Points

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

y = x² − 32√x

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First, identify the domain of the function y = x² − 32√x. Since the square root function √x is only defined for x ≥ 0, the domain of the function is x ≥ 0.

Next, find the derivative of the function y with respect to x. The derivative of y = x² is 2x, and the derivative of −32√x is −16/√x. Therefore, the derivative y' is y' = 2x − 16/√x.

Set the derivative y' equal to zero to find the critical points: 2x − 16/√x = 0. Solve this equation for x to find the critical points.

To solve 2x − 16/√x = 0, multiply through by √x to eliminate the fraction: 2x√x = 16. Then, solve for x by isolating x on one side of the equation.

Finally, evaluate the function at the endpoints of the domain and at any critical points found to determine the behavior of the function at these points. The endpoint in this case is x = 0, and any critical points found from the previous step.

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are important because they can indicate local maxima, minima, or points of inflection. To find them, take the derivative of the function and solve for the values of x where the derivative equals zero or does not exist.

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input. It is a fundamental tool in calculus for analyzing the behavior of functions. For the function y = x² − 32√x, the derivative is found using the power rule and the chain rule, resulting in y' = 2x - 16x^(-1/2).

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For y = x² − 32√x, the domain is restricted to x ≥ 0 because the square root function is only defined for non-negative numbers. Identifying the domain is crucial for determining endpoints and ensuring the function is evaluated correctly.

Finding Critical PointsIn Exercises 41–50, determine all critical points and all domain endpoints for each function.y = x² − 32√x