Question

analyze the following graph of $f(x)$. select all of the intervals over which the function $f(x)$ is concave up.
select all that apply:
$(-\infty,0)$
$(0,\frac{1}{2})$
$(\frac{1}{2},\infty)$
none of the above.

Explanation:

Step1: Recall the concavity rule

A function \(y = f(x)\) is concave up when \(f''(x)>0\). Since \(f''(x)\) is the derivative of \(f'(x)\), we need to find where \(f'(x)\) is increasing.

Step2: Analyze the graph of \(f'(x)\)

Looking at the graph of \(y = f'(x)\), we observe the slope of \(f'(x)\). The slope of \(f'(x)\) (which is \(f''(x)\)) is negative everywhere on the given graph.

Answer:

None of the above.