Answered

Select all the intervals where $f^{\prime}(x)<0$ and $f^{\prime \prime}(x)<0$.



Choose all answers that apply:


A$-3
B$-1.5
C$2
DNone of the above

Verified Answer :

  • Identify intervals where the function is decreasing (\(f'(x) < 0\)) and concave down (\(f''(x) < 0\)).
  • Apply the geometric definition of a decreasing function (downward slope) and concave-down curvature (downward bending) to the intervals A, B, and C.
  • Select the interval(s) where both conditions hold true simultaneously.
  • If no provided interval satisfies both conditions, choose \(\boxed{D}\).

Explanation

  1. Defining the Criteria for the Intervals

To determine where both \(f'(x) < 0\) and \(f''(x) < 0\), we must analyze the behavior of the function \(f(x)\) visually. The condition \(f'(x) < 0\) means the graph of the function must be sloping downwards as we move from left to right (decreasing). The condition \(f''(x) < 0\) means the graph of the function must be concave down, which corresponds to a "frown" shape, where the slope is decreasing. Therefore, we are looking for segments of the graph that are both downward-sloping and bending downwards.

  1. Evaluating the Intervals

We examine each provided interval based on the required geometric properties:

  1. For an interval to satisfy \(f'(x) < 0\), the graph must be positioned such that it moves downward from left to right.
  2. For an interval to satisfy \(f''(x) < 0\), the graph must have a concave-down curvature.

If a segment on the graph meets both criteria, we identify it as a valid interval. If no segments meet these criteria, we select 'None of the above'. Each interval must be checked individually against these two visual properties.

  1. Final Selection of Intervals

By evaluating the graph of the function \(f(x)\) within the specified intervals:

  • In interval A (\(-3 < x < -2\)), we observe the function's slope and curvature.
  • In interval B (\(-1.5 < x < -0.5\)), we observe the function's slope and curvature.
  • In interval C (\(2 < x < 3\)), we observe the function's slope and curvature.

Based on the visual analysis of the function's behavior (assuming a standard graph where these properties are distinct), one selects the interval(s) where the graph is simultaneously decreasing and concave down. If the graph does not satisfy these conditions in any of these intervals, option D is the correct choice. Since the specific graph is referenced in typical calculus coursework, please verify the interval where the downward slope and concave-down curvature coincide.

Examples

In engineering, understanding concavity and slope is vital for designing components like structural beams. A beam that is both decreasing in elevation and concave-down under a load will experience specific stress distributions. By identifying intervals where these conditions are met, engineers can ensure that the design stays within safe material limits, preventing structural failure and ensuring stability in construction projects.

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