3. Find the largest rectangle (that is , the rectangle with largest area) that fits inside the graph of the parabola y=x^2 below the line y=a (a is an unspecified constant value), with the top side of the rectangle on the horizontal line y=a

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Answer 1

To find the largest rectangle that fits inside the graph of the parabola

below the line

, we need to maximize the area of the rectangle.Let's denote the x-coordinate of the vertex of the rectangle as

. Since the rectangle is symmetric about the y-axis, its top side will be on the horizontal line

and its bottom side will be on the parabola

. Therefore, the height of the rectangle is

.The width of the rectangle is twice the distance from the y-axis to the x-coordinate of the vertex, which is

.The area of the rectangle is given by the product of its width and height, which is

.To maximize the area, we need to find the value of

that maximizes the function

. We can do this by taking the derivative of

with respect to

and setting it equal to zero:

Solving for

, we get

.Therefore, the largest rectangle that fits inside the graph of the parabola

below the line

has a width of

and a height of

.

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