Question

analyze the following graph of ( f^{prime}(x) ). select all of the intervals over which the graph of ( f(x) ) is concave up.
select all that apply:
( (-infty,-1) )
( (-1,1) )
( (1, infty) )
none of the above,

Explanation:

Step1: Recall the concavity rule

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

Step2: Analyze the slope of \(y = f'(x)\)

• For the interval \((-\infty,-1)\): The slope of \(y = f'(x)\) (i.e., \(f''(x)\)) is positive.
• For the interval \((-1,1)\): The slope of \(y = f'(x)\) (i.e., \(f''(x)\)) is negative.
• For the interval \((1,\infty)\): The slope of \(y = f'(x)\) (i.e., \(f''(x)\)) is positive.

Answer:

\((-\infty,-1)\) and \((1,\infty)\)