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

f(x) = x²/₃ (x²-4)

검증된 단계별 안내

First, find the derivative of the function f(x) = x^(2/3) * (x^2 - 4). Use the product rule for differentiation, which states that if you have a function h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x).

Let u(x) = x^(2/3) and v(x) = x^2 - 4. Find the derivatives u'(x) and v'(x). For u(x) = x^(2/3), use the power rule to get u'(x) = (2/3)x^(-1/3). For v(x) = x^2 - 4, the derivative is v'(x) = 2x.

Apply the product rule: f'(x) = u'(x) * v(x) + u(x) * v'(x). Substitute the derivatives and original functions: f'(x) = (2/3)x^(-1/3) * (x^2 - 4) + x^(2/3) * 2x.

Simplify the expression for f'(x). Combine like terms and simplify the expression to make it easier to analyze the sign of f'(x).

Determine the intervals where f'(x) > 0 (function is increasing) and f'(x) < 0 (function is decreasing). Find the critical points by setting f'(x) = 0 and solving for x. Use these critical points to test intervals on the number line to determine where the function is increasing or decreasing.

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주요 개념

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Critical Points

Critical points are values of x in 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.

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 those intervals. 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.

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 graphing and optimization problems.

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.f(x) = x²/₃ (x²-4)