A function has its first derivative graphed below.

Based on the graph of above, on which interval(s) is concave down? Select all that apply.

Question

A function  has its first derivative graphed below.

Based on the graph of  above, on which interval(s) is  concave down? Select all that apply.

BlackTom AI · Accepted Answer

The task asks to determine where f''(x) < 0 (concave down) based on the graph of f'(x). The key idea is: - Concave down corresponds to f''(x) < 0, which happens where f'(x) is decreasing (the slope of f' is negative). - Concave up corresponds to f''(x) > 0, which happens where f'(x) is increasing (the slope of f' is positive). Step-by-step analysis approach (independent of the exact numbers): - Look at the portions of the f'(x) graph where the curve is sloping downward (the y-values of f' are getting smaller as x increases). Those x-intervals are where f''(x) < 0, i.e., concave down. - Conversely, where the f'(x) curve slopes upward, f''(x) > 0, i.e., concave up. Given the graph shown: - From the far left toward the first visible turning point (the leftmost valley of f'(x)), the graph of f' is decreasing (it goes from a high value down to a low value). Therefore, on that interval f''(x) < 0 (concave down). - From that valley up to the next turning point (the subsequent peak of f'), f' is increasing. Therefore, f''(x) > 0 (concave up) on that interval. - From that peak down to the next valley of f', f' is decreasing again. Therefore, concave down on that interval. - After that valley, as f'(x) increases steeply toward the right end, f' is increasing, so the function becomes concave up on that final interval. Important note about the provided data: - The problem statement mentions multiple answer choices for intervals where f is concave down, but the actual answer options are not included in the provided data. Without the explicit interval options, I cannot enumerate exactly which interval labels (e.g., (a,b), (c,d), or specific x-values) qualify. What I would need to complete this precisely: - The list of interval options for concave down (the answer choices). - If available, the x-coordinates of the key turning points on the f'(x) graph (where f' changes from decreasing to increasing or vice versa) to mark the concave down regions exactly. If you provide the missing answer options (the specific intervals), I can state exactly which of them correspond to concave down and explain why each is or isn’t correct using the increasing/decreasing nature of f'(x) on those intervals.

A function has its first derivative graphed below. Based on the graph of above, on which interval(s) is concave down? Select all that apply.

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