Final answer:
The function y= 3-2x² has a second derivative of -4, which is negative and indicates the function is concave down on all real numbers. There are no points of inflection as the concavity does not change.
Step-by-step explanation:
To determine concavity and the x-values where points of inflection occur for the function y= 3-2x², we look at the second derivative of the function. The second derivative of y with respect to x is y'' = -4, which is constant and negative. Since the second derivative is negative for all values of x, the function is concave down on the entire real number line.
Because the concavity does not change (the second derivative does not switch signs), there are no points of inflection. Points of inflection occur where the concavity changes — that is, where the second derivative is zero or undefined, and this is not the case for this function.
Answer to Part A: The function is concave down on all real numbers.
Answer to Part B: The function is never concave up; therefore, it is always concave down.