2. sketch a single graph with the characteristics described below:\n● concave up and decreasing on the…

2. sketch a single graph with the characteristics described below:\n● concave up and decreasing on the interval (-∞, -2)\n● concave up and increasing on the interval (-2, 3)\n● concave down and increasing on the interval (3,6)\n● concave down and decreasing on the interval (6, ∞)

2. sketch a single graph with the characteristics described below:\n● concave up and decreasing on the interval (-∞, -2)\n● concave up and increasing on the interval (-2, 3)\n● concave down and increasing on the interval (3,6)\n● concave down and decreasing on the interval (6, ∞)

Answer

Explanation:

Step1: Analyze concavity and monotonicity

  • For ( (-\infty,-2) ): Concave up (( f''>0 )) and decreasing (( f'<0 )). So the slope of the tangent line is negative and the rate of change of the slope (concavity) is positive.
  • For ( (-2,3) ): Concave up (( f''>0 )) and increasing (( f'>0 )). The slope of the tangent line is positive and the rate of change of the slope (concavity) is positive.
  • For ( (3,6) ): Concave down (( f''<0 )) and increasing (( f'>0 )). The slope of the tangent line is positive and the rate of change of the slope (concavity) is negative.
  • For ( (6,\infty) ): Concave down (( f''<0 )) and decreasing (( f'<0 )). The slope of the tangent line is negative and the rate of change of the slope (concavity) is negative.

Step2: Sketch the graph

  • Start from the left (( x =-\infty )). Since it is concave up and decreasing on ( (-\infty,-2) ), draw a curve that is bending upwards (like a cup) and going down as ( x ) increases towards ( - 2 ).
  • At ( x=-2 ), the function changes from decreasing to increasing. Since it is still concave up on ( (-2,3) ), draw a curve that is bending upwards (like a cup) and going up as ( x ) increases towards ( 3 ).
  • At ( x = 3 ), the function changes concavity from up to down. Since it is still increasing on ( (3,6) ), draw a curve that is bending downwards (like a cap) and going up as ( x ) increases towards ( 6 ).
  • At ( x=6 ), the function changes from increasing to decreasing. Since it is concave down on ( (6,\infty) ), draw a curve that is bending downwards (like a cap) and going down as ( x ) increases.

Answer:

A possible graph is drawn according to the above - described steps. (Since this is a sketching problem, a hand - drawn or computer - generated graph that meets the concavity and monotonicity conditions on the given intervals is the answer. For example, a polynomial - like function ( y=-\frac{1}{12}x^{4}+\frac{1}{6}x^{3}+\frac{3}{2}x^{2}- 6x) can be used as a reference for sketching. First, ( y'=- \frac{1}{3}x^{3}+\frac{1}{2}x^{2}+3x - 6=-\frac{1}{6}(2x^{3}-3x^{2}-18x + 36)=-\frac{1}{6}(x - 3)(2x^{2}+3x - 12)). ( y''=-x^{2}+x + 3=-(x^{2}-x - 3)). By analyzing the signs of ( y') and ( y'') on the intervals ( (-\infty,-2)), ( (-2,3)), ( (3,6)) and ( (6,\infty)) we can confirm the concavity and monotonicity properties.)