A rectangle is inscribed into the region bounded by the graph of f(x)=(x^2-1)^2 and the x-axis, in such a way that one side of the rectangle lies on the x-axis and the two vertices lie on the graph of f(x). what is the maximum possible area of such a rectangle?
Please help I have no idea how to solve this problem.
Your graph looks like a W and I assume you want the rectangle placed inside the centre part.
Let the two other vertices be (x,y) and (-x,y)
So the length is 2x and the height is y
Area = 2xy = 2x(x^4 - 2x^2 + 1)
= 2x^5 - 4x^3 + 2x
d(area)/dx = 10x^4 - 12x^2 + 2
= 0
5x^4 - 6x^2 + 1 = 0
treating this as a quadratic in x^2
x^2 - 1)(5x^2 - 1) = 0
x^2 = 1 or x^2 = 1/5
x = ±1 or x = ±1/√5
clearly, if x = 1, area = 0 , must be the minimum
and
when x = 1/√5
area = 2x(x^2 - 1)^2
= 2/√5(-4/5)^2
= 2/√5 (16/25)
or appr .572 units^2
check my arithmetic
To find the maximum possible area of the rectangle inscribed in the region bounded by the given graph, we need to follow these steps:
1. Understand the problem: We are given a function, f(x) = (x^2-1)^2, and we need to inscribe a rectangle within the region bounded by the graph of this function and the x-axis. The rectangle should have one side on the x-axis and two vertices on the graph of f(x).
2. Visualize the problem: Draw the graph of the function f(x) = (x^2-1)^2. This will help you understand the shape of the region and the potential position of the rectangle within it.
3. Determine the variables: Let's assume that the width of the rectangle is "x". Since the rectangle has one side on the x-axis, the length of the rectangle will be determined by the height of the graph of f(x) at two points.
4. Express the length of the rectangle: The length of the rectangle will be the difference between the values of f(x) at two points. We need to find these two points.
5. Find the points of tangency: To find the two points on the graph of f(x) where the vertices of the rectangle lie, we need to determine where the slope of the tangent line to the graph is zero. The points of tangency will be the x-values where f'(x) = 0.
6. Calculate the length of the rectangle: Find the values of f(x) at the two points of tangency, and subtract them to determine the length of the rectangle.
7. Calculate the area: Multiply the length and width of the rectangle to find its area.
8. Determine the maximum area: Use calculus to find the maximum value of the area by taking the derivative of the area function with respect to the variable x, setting it equal to zero, and solving for x.
9. Solve for x and plug it back into the area formula: Find the value of x that maximizes the area of the rectangle. Then, substitute this value back into the area formula to obtain the maximum area.
By following these steps, you can find the maximum possible area of the rectangle inscribed in the given region.