Final answer:
The open intervals on which the graph of the function f(x) = (x²)/(x² + 64) is concave upward or concave downward are (-∞, -8), (-8, 8), and (8, +∞).
Step-by-step explanation:
To determine the open intervals on which the graph of the function is concave upward or concave downward, we need to find the second derivative of the function. Let's start by finding the first and second derivatives of the function f(x) = (x²)/(x² + 64).
The first derivative of f(x) is f'(x) = (2x(x² + 64) - x²(2x))/(x² + 64)². Simplifying this expression, we get f'(x) = 128x/(x² + 64)² - 2x³/(x² + 64)².
The second derivative of f(x) is f''(x) = (128(x² + 64)² - 256x²(x² + 64))/(x² + 64)⁴. Simplifying this expression, we get f''(x) = (8192 - 192x^2)/(x^2 + 64)⁴.
To determine the concavity of the graph at a given x-value, we need to analyze the sign of f''(x). If f''(x) is positive, the graph is concave upward; if f''(x) is negative, the graph is concave downward.
Now, let's find the x-values where f''(x) changes sign, which will give us the open intervals of concavity. Setting f''(x) equal to zero, we have (8192 - 192x^2)/(x^2 + 64)⁴ = 0. Solving this equation, we get x = ±8. So, the intervals of concavity are (-∞, -8), (-8, 8), and (8, +∞). In the interval (-∞, -8) and (8, +∞), the graph is concave upward, and in the interval (-8, 8), the graph is concave downward.