Final answer:
The function f(x) = ln(8x² + 1) does not have any critical points or inflection points.
Step-by-step explanation:
To find the critical points of the function f(x) = ln(8x² + 1), we need to find the values of x where the derivative of f(x) is equal to zero or undefined.
Taking the derivative of f(x) with respect to x, we get f'(x) = 16x / (8x² + 1).
Setting f'(x) equal to zero and solving for x, we find that there is a critical point at x = 0. When x = 0, the function is undefined, so this is an asymptote rather than a critical point. Therefore, there are no critical points for the function f(x) = ln(8x² + 1).
Inflection points occur where the second derivative of the function changes sign. To find the inflection points, we need to take the second derivative of f(x). The second derivative of f(x) = ln(8x² + 1) is given by f''(x) = (128x² - 16) / (8x² + 1)².
Setting f''(x) equal to zero and solving for x, we find that there are no real solutions. Therefore, the function f(x) = ln(8x² + 1) has no inflection points.