Question

For what intervals is g(x)= 1/ x² +1 concave down?

a) (−[infinity],[infinity])
b) (−[infinity],−1)∪(1,[infinity])
c) (0,[infinity])
d)(0, ,[infinity])

Answer 1

The function g(x) = 1/(x²+1) is not concave down on any interval since its second derivative is always positive. The provided answer options do not accurately reflect the concavity of the function, indicating an error in the options.

Step-by-step explanation:

The question refers to determining the intervals where the function g(x) = 1/(x²+1) is concave down. To find the concavity of a function, we look at the second derivative of the function, g''(x). The function is concave down where g''(x) is less than zero. After differentiating the function twice, we find that the second derivative is always positive, which means that g(x) is never concave down. Therefore, the correct answer is that g(x) is not concave down on any interval, which is not listed as an option in the given choices. Thus, there seems to be a mistake in the question as none of the provided options correctly describe the concavity of g(x).