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In probability density functions (PDF), the non-negativity condition assures us that the function is always zero or positive. This reflects the idea that probabilities cannot be negative. For a function like \( f(x) \), defined for certain intervals, checking for non-negativity involves finding where \( C(2x - x^3) \) is greater than or equal to zero.

By solving the equation \( C(2x - x^3) = 0 \), we determine critical points. Critical points occur where the function changes from positive to negative or vice versa. For instance, solving \( x(2-x^2) = 0 \) gives potential points where the sign of \( f(x) \) could change.

Remember: Only non-negative intervals matter for PDFs. So if our critical points are \( x = 0 \) and \( x = \sqrt{2} \), we focus on \( 0 < x < \sqrt{2} \), ensuring \( f(x) \) stays non-negative.

For a function to qualify as a probability density function, the integral of the function across its whole range must equal one. Think of it as summing all the possibilities - they should total one full probability.

For the example function \( f(x) = C(2x - x^3) \), its range for non-zero values is \( 0 < x < \sqrt{2} \). Thus, we calculate:

\[ \int_{0}^{\sqrt{2}} C(2x - x^3) dx = 1 \]

This equation ensures that the total area under the PDF curve sums up to one, providing a solid basis for describing probabilities continuously over an interval.

Integral calculation is a fundamental process in confirming whether a function can serve as a valid probability density function. It involves finding the antiderivative and evaluating it over a specific interval.

Consider the function \( C(2x - x^3) \). We break down the integral:
\[ C \left[ \int_{0}^{\sqrt{2}} 2x \ dx - \int_{0}^{\sqrt{2}} x^3 \ dx \right] \]

Each integral is solved separately. For \( \int_{0}^{\sqrt{2}} 2x \ dx \), the antiderivative is \( x^2 \), evaluated from 0 to \(\sqrt{2}\). For \( \int_{0}^{\sqrt{2}} x^3 \ dx \), the antiderivative is \( \frac{1}{4}x^4 \), also evaluated from 0 to \( \sqrt{2} \). Combining these evaluative steps yields a full solution, leading to finding the constant \( C \).

Critical points are where the derivative of a function is zero or undefined; it helps determine changes in direction or slope of a function. These points can indicate maximums, minimums, or points of inflection.

In the context of probability, finding critical points is crucial when determining intervals of non-negativity. For example, when finding critical points for \( C(2x - x^2) \), solve \( 2x - x^2 = 0 \) yielding \( x = 0 \) or \( x = 2 \).

• This tells us where the function either "flattens out" or could change from increasing to decreasing.
• Assistance in designating valid intervals for the PDF, ensuring \( f(x) \) remains non-negative in \( 0 < x < 2 \).