9. suppose that the graph above is ( y = v(x) ). find:\na. interval(s) where ( v(x) ) is increasing:______dec…

9. suppose that the graph above is ( y = v(x) ). find:\na. interval(s) where ( v(x) ) is increasing:______decreasing:______\nb. ( x )-coordinate(s) where ( v(x) ) has a local max:______local min:______\nc. interval(s) where ( v(x) ) is concave up:______concave down:______\nd. ( x )-coordinate(s) where ( v(x) ) has an inflection point:______\n10. suppose that the graph at the top of the page is ( y = v(x) ). find:\na. interval(s) where ( v(x) ) is increasing:______decreasing:______\nb. ( x )-coordinate(s) where ( v(x) ) has a local max:______local min:______\nc. interval(s) where ( v(x) ) is concave up:______concave down:______\nd. ( x )-coordinate(s) where ( v(x) ) has an inflection point:______

9. suppose that the graph above is ( y = v(x) ). find:\na. interval(s) where ( v(x) ) is increasing:______decreasing:______\nb. ( x )-coordinate(s) where ( v(x) ) has a local max:______local min:______\nc. interval(s) where ( v(x) ) is concave up:______concave down:______\nd. ( x )-coordinate(s) where ( v(x) ) has an inflection point:______\n10. suppose that the graph at the top of the page is ( y = v(x) ). find:\na. interval(s) where ( v(x) ) is increasing:______decreasing:______\nb. ( x )-coordinate(s) where ( v(x) ) has a local max:______local min:______\nc. interval(s) where ( v(x) ) is concave up:______concave down:______\nd. ( x )-coordinate(s) where ( v(x) ) has an inflection point:______

Answer

Explanation:

Step1: Analyze (v(x)) (Question 9)

  • Increasing/Decreasing: A function (y = v(x)) is increasing when the slope is positive ((v^{\prime}(x)>0)) and decreasing when the slope is negative ((v^{\prime}(x)<0)). From the graph, (v(x)) is increasing on ((-\infty,1)) and decreasing on ((1,\infty)).
  • Local Max/Min: A local maximum occurs where the function changes from increasing to decreasing. So (x = 1) is a local max. There is no local min as the function does not change from decreasing to increasing.
  • Concavity: Concave - up means (v^{\prime\prime}(x)>0) (the slope of (v^{\prime}(x)) is increasing) and concave - down means (v^{\prime\prime}(x)<0) (the slope of (v^{\prime}(x)) is decreasing). The graph of (v(x)) is concave - down everywhere (((-\infty,\infty))) as it has a "frown" shape. There is no concave - up interval.
  • Inflection Point: An inflection point is where the concavity changes. Since the concavity does not change, there is no inflection point.

Step2: Analyze (v^{\prime}(x)) (Question 10)

  • Increasing/Decreasing of (v(x)): (v(x)) is increasing when (v^{\prime}(x)>0). From the graph (assuming (y = v^{\prime}(x))), (v^{\prime}(x)>0) on ((-1,3)) and (v^{\prime}(x)<0) on ((-\infty,-1)\cup(3,\infty)). So (v(x)) is increasing on ((-1,3)) and decreasing on ((-\infty,-1)\cup(3,\infty)).
  • Local Max/Min of (v(x)): (v(x)) has a local maximum when (v^{\prime}(x)) changes from positive to negative ((x = 3)) and a local minimum when (v^{\prime}(x)) changes from negative to positive ((x=-1)).
  • Concavity of (v(x)): (v(x)) is concave - up when (v^{\prime\prime}(x)= (v^{\prime}(x))^{\prime}>0) (the slope of (v^{\prime}(x)) is positive). The slope of (v^{\prime}(x)) is positive on ((-\infty,1)) and negative on ((1,\infty)). So (v(x)) is concave - up on ((-\infty,1)) and concave - down on ((1,\infty)).
  • Inflection Point of (v(x)): An inflection point occurs where (v^{\prime\prime}(x) = 0) (where the slope of (v^{\prime}(x)) changes sign). So (x = 1) is the inflection point.

Answer:

Question 9 A. Increasing: ((-\infty,1)); Decreasing: ((1,\infty)) B. Local max: (x = 1); Local min: None C. Concave up: None; Concave down: ((-\infty,\infty)) D. Inflection point: None

Question 10 A. Increasing: ((-1,3)); Decreasing: ((-\infty,-1)\cup(3,\infty)) B. Local max: (x = 3); Local min: (x=-1) C. Concave up: ((-\infty,1)); Concave down: ((1,\infty)) D. Inflection point: (x = 1)