Let B, C and D be fixed points and let A be a moveable point such that line segment AB is parallel to line segment CD. Let a be the distance between C and D and let h be the distance between AB and CD. Find x, the distance between A and B, such that the sum of the areas of $\displaystyle \Delta$ ABE and $\displaystyle \Delta$ CDE is minimized. You may move the slider in the figure below to see the effect of moving point A.

(a) Let y equal the altitude of $\displaystyle \Delta$ ABE, measured from E to line segment AB. Write the function $\displaystyle {A}{\left({x}\right)}$ that gives the sum of the areas of $\displaystyle \Delta$ ABE and $\displaystyle \Delta$ CDE. Make sure your function is written in terms of $\displaystyle {a}$ , $\displaystyle {h}$ , $\displaystyle {x}$ , and not $\displaystyle {y}$ .

      $\displaystyle {A}{\left({x}\right)}=$ Preview Question 6 Part 1 of 7  

(b) What is the open interval on which $\displaystyle {A}{\left({x}\right)}$ is defined?

      Preview Question 6 Part 2 of 7  

(c) $\displaystyle {A}'{\left({x}\right)}=$ Preview Question 6 Part 3 of 7  

(d) What is the critical number for $\displaystyle {A}{\left({x}\right)}$ that lies in the interval identified in part (b)? Note: your answer should be expressed in terms of a.

      Preview Question 6 Part 4 of 7  

(e) Evaluate $\displaystyle {A}{''}{\left({c}\right)}$ where c is the critical number found in part (d). Note: your answer should be expressed in terms of a and h.

      $\displaystyle {A}{''}{\left({c}\right)}=$ Preview Question 6 Part 5 of 7  

(f) Since $\displaystyle {A}{''}$ is Select an answer positive negative for this c value, $\displaystyle {A}{\left({x}\right)}$ has a Select an answer minimum maximum by the second derivative test for relative extrema.