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 `Delta` ABE and `Delta` CDE is minimized. You may move the slider in the figure below to see the effect of moving point A.

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

`A(x)=`

(b) What is the open interval on which `A(x)` is defined?

(c) `A'(x)=`

(d) What is the critical number for `A(x)` that lies in the interval identified in part (b)? Note: your answer should be expressed in terms of a.

(e) Evaluate `A''(c)` where c is the critical number found in part (d). Note: your answer should be expressed in terms of a and h.

`A''(c)=`

(f) Since `A''` is for this c value, `A(x)` has a by the second derivative test for relative extrema.