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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.3.36
Chapter 4, Problem 4.3.36

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = x²√(9 - x²) on (-3,3)

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First, find the derivative of the function f(x) = x²√(9 - x²). Use the product rule for differentiation, as the function is a product of x² and √(9 - x²).
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, let u(x) = x² and v(x) = √(9 - x²).
Differentiate u(x) = x² to get u'(x) = 2x. For v(x) = √(9 - x²), use the chain rule: v'(x) = (1/2)(9 - x²)^(-1/2) * (-2x) = -x/(√(9 - x²)).
Substitute u'(x), v(x), u(x), and v'(x) into the product rule: f'(x) = 2x√(9 - x²) + x²(-x/√(9 - x²)). Simplify this expression to find f'(x).
Determine where f'(x) is positive or negative to find the intervals of increase or decrease. Set f'(x) > 0 for increasing intervals and f'(x) < 0 for decreasing intervals. Solve these inequalities considering the domain (-3, 3).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Critical Points

Critical points are values of x in the domain of a function where the derivative is either zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. To find critical points, we first compute the derivative of the function and set it equal to zero, solving for x.
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First Derivative Test

The First Derivative Test is a method used to determine the behavior of a function at its critical points. By analyzing the sign of the derivative before and after each critical point, we can conclude whether the function is increasing or decreasing in the intervals defined by these points. If the derivative changes from positive to negative, the function has a local maximum; if it changes from negative to positive, it has a local minimum.
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Intervals of Increase and Decrease

Intervals of increase and decrease refer to the ranges of x-values where a function is respectively rising or falling. A function is increasing on an interval if its derivative is positive throughout that interval, and decreasing if the derivative is negative. Identifying these intervals helps in understanding the overall behavior of the function and is crucial for sketching its graph.
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