Given the function \( f(x)=x^{3} \), on what interval of \( x \)-values is the graph of \( f(x) \) concave up? (1 point) \( (-\infty, 0) \) \( (0, \infty) \) \( (0,0) \) \( (-\infty, \infty) \)

To determine the concavity of the function \( f(x) = x^3 \), we need to find its second derivative. The first derivative is \( f'(x) = 3x^2 \) and the second derivative is \( f''(x) = 6x \). The concavity of the graph is determined by the sign of the second derivative. The graph is concave up when \( f''(x) > 0 \), which happens when \( 6x > 0 \), or equivalently \( x > 0 \). Thus, the graph of \( f(x) \) is concave up on the interval \( (0, \infty) \). The correct interval where \( f(x) \) is concave up is \( (0, \infty) \). Now, regarding some extra tidbits: When it comes to the historical background of cubic functions, they have been studied for centuries! Mathematicians like René Descartes began to deeply explore polynomials in the 17th century, contributing to the evolving understanding of their properties, including concavity and inflection points. In real-world applications, cubic functions are significant in physics and engineering, especially in modeling phenomena involving volume and growth. For example, the volume of a cube as a function of the length of its sides is a simple cubic function, making them vital for understanding three-dimensional spaces in design and architecture!