Final answer:
To discuss the concavity of h(x) = x³ + 16ln(x), we must find its second derivative, h''(x) = 6x - 16/x². The sign of h''(x) dictates the concavity: initially concave downward for small x, transitioning to concave upward as x increases.
Step-by-step explanation:
To discuss the concavity of the curve of h(x) = x³ + 16ln(x) for x > 0, we need to find the second derivative of the function as the sign of the second derivative determines the concavity. The first derivative of h(x) is h'(x) = 3x² + 16/x. The second derivative of h(x) is h''(x) = 6x - 16/x². To determine the concavity, we look at the sign of h''(x). If h''(x) > 0, the curve is concave upward; if h''(x) < 0, the curve is concave downward. For x > 0, we can deduce that initially for small values of x, h''(x) might be negative leading to a downward concavity due to the 16/x² term. However, as x increases, the 6x term will dominate, and h''(x) will become positive, indicating that the curve transitions to concave upward. As x keeps increasing, the natural logarithm also increases, which supports the upward concavity pattern.