watermelon , if area of the based is A=2X2+10 find the length and breadth of the rectangular base
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Answer:
Let  denote the length of the side of the garden perpendicular to the rock wall and  denote the length of the side parallel to the rock wall. Then the area of the garden is

We want to find the maximum possible area subject to the constraint that the total fencing is  From (Figure), the total amount of fencing used will be  Therefore, the constraint equation is

Solving this equation for  we have  Thus, we can write the area as

Before trying to maximize the area function  we need to determine the domain under consideration. To construct a rectangular garden, we certainly need the lengths of both sides to be positive. Therefore, we need  and  Since  if  then  Therefore, we are trying to determine the maximum value of  for  over the open interval  We do not know that a function necessarily has a maximum value over an open interval. However, we do know that a continuous function has an absolute maximum (and absolute minimum) over a closed interval. Therefore, let’s consider the function  over the closed interval  If the maximum value occurs at an interior point, then we have found the value  in the open interval  that maximizes the area of the garden. Therefore, we consider the following problem:
Maximize  over the interval 
As mentioned earlier, since  is a continuous function on a closed, bounded interval, by the extreme value theorem, it has a maximum and a minimum. These extreme values occur either at endpoints or critical points. At the endpoints,  Since the area is positive for all  in the open interval  the maximum must occur at a critical point. Differentiating the function  we obtain

Therefore, the only critical point is  ((Figure)). We conclude that the maximum area must occur when  Then we have  To maximize the area of the garden, let  ft and  The area of this garden is 
Figure 2. To maximize the area of the garden, we need to find the maximum value of the function