Finding Critical Points

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

f(x) = x(4 − x)³

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First, understand that critical points occur where the derivative of the function is zero or undefined. Begin by finding the derivative of the function f(x) = x(4 - x)^3.

Apply the product rule to differentiate f(x) = x(4 - x)^3. The product rule states that if you have a function h(x) = u(x)v(x), then h'(x) = u'(x)v(x) + u(x)v'(x). Here, u(x) = x and v(x) = (4 - x)^3.

Differentiate u(x) = x to get u'(x) = 1. Next, differentiate v(x) = (4 - x)^3 using the chain rule. The chain rule states that if you have a composite function g(h(x)), then the derivative is g'(h(x)) * h'(x). Let g(y) = y^3 and h(x) = 4 - x, then g'(y) = 3y^2 and h'(x) = -1.

Combine the results from the product and chain rules to find f'(x). Substitute u'(x), v(x), u(x), and v'(x) into the product rule formula: f'(x) = 1 * (4 - x)^3 + x * 3(4 - x)^2 * (-1). Simplify this expression.

Set the derivative f'(x) equal to zero and solve for x to find the critical points. Also, consider the endpoints of the domain of f(x), which are determined by the context of the problem or any given interval.

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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 critical points, 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 f(x) = x(4 − x)³, use the product rule and chain rule to find its derivative, which is essential for identifying critical points.

Domain Endpoints

Domain endpoints are the boundary values of the domain of a function, where the function is defined. These points are crucial when analyzing a function's behavior over its entire domain, especially when determining absolute extrema. For polynomial functions like f(x) = x(4 − x)³, the domain is typically all real numbers, but endpoints are considered in restricted domains.

Finding Critical PointsIn Exercises 41–50, determine all critical points and all domain endpoints for each function.f(x) = x(4 − x)³