analyze the following graph of ( f^{prime}(x) ). select all of the intervals over which the graph of ( f(x)…

analyze the following graph of ( f^{prime}(x) ). select all of the intervals over which the graph of ( f(x) ) is concave up.\nselect all that apply:\n( (-infty,-1) )\n( (-1,1) )\n( (1, infty) )\nnone of the above,

analyze the following graph of ( f^{prime}(x) ). select all of the intervals over which the graph of ( f(x) ) is concave up.\nselect all that apply:\n( (-infty,-1) )\n( (-1,1) )\n( (1, infty) )\nnone of the above,

Answer

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))