calculus question!

Question

Let f(x)=1/2x^2+3. Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inflection points of f.

1.	f is concave up on the intervals
2.	f is concave down on the intervals
3.	The inflection points occur at x =

Answer 1

To find the intervals on which f(x) is concave up/down, we need to calculate the second derivative of f(x):

f(x) = 1/2x^2 + 3

f'(x) = x

f''(x) = 1

Since f''(x) is positive for all x, f(x) is always concave up.

To find the inflection points of f(x), we need to solve the equation f''(x) = 0. However, since f''(x) is a constant function, it never equals zero. Therefore, f(x) has no inflection points.

In summary, the function f(x) = 1/2x^2 + 3 is always concave up and has no inflection points.