Find the maximum area of a rectangle inscribed between the x-axis and f(x)=-x^4+3

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To find the maximum area of a rectangle inscribed between the x-axis and the curve f(x) = -x^4 + 3, we need to determine the dimensions of the rectangle that will yield the largest possible area.

Step 1: Determine the x-coordinate(s) where the curve intersects the x-axis.
To find the x-coordinate(s) where the curve intersects the x-axis, we set f(x) = 0 and solve for x.
- x^4 + 3 = 0

Step 2: Solve for x.
Using algebraic methods or a graphing calculator, we can find that there are two real solutions for x:
x = -1 and x = 1.

Step 3: Determine the y-coordinates corresponding to these x-values.
To find the y-coordinate(s) corresponding to the x-values -1 and 1, we substitute these values into the equation f(x) = -x^4 + 3.
For x = -1:
f(-1) = -(-1)^4 + 3 = -1 + 3 = 2

For x = 1:
f(1) = -(1)^4 + 3 = -1 + 3 = 2

Step 4: Calculate the length and width of the rectangle.
Since the rectangle is inscribed between the x-axis and the curve, the length of the rectangle will be twice the distance between the x-intercepts:
Length = 2 * distance between x-intercepts = 2 * (1 - (-1)) = 4

The width of the rectangle will be the y-coordinate of the curve at the x-intercept(s):
Width = y-coordinate at x-intercept = 2

Step 5: Calculate the area of the rectangle.
The area of the rectangle is given by the formula: Area = Length * Width.

Area = 4 * 2 = 8

Therefore, the maximum area of a rectangle inscribed between the x-axis and the curve f(x) = -x^4 + 3 is 8 square units.

To find the maximum area of a rectangle inscribed between the x-axis and the graph of a function, you need to determine the dimensions (width and height) of the rectangle with the maximum area.

In this case, the function is f(x) = -x^4 + 3.

Step 1: Determine the x-values where the function intersects the x-axis.
To find these values, set f(x) equal to zero and solve for x:
0 = -x^4 + 3
x^4 = 3
x = ±√3 or approximately ±1.732 (rounded to 3 decimal places).

Step 2: Determine the corresponding y-values on the graph of the function.
Substitute the x-values obtained in step 1 into the function f(x) to find the corresponding y-values:
f(√3) = -(√3)^4 + 3 = -3 + 3 = 0
f(-√3) = -(-√3)^4 + 3 = -3 + 3 = 0

Step 3: Calculate the width and height of the rectangle.
The width of the rectangle is given by the difference between the x-values on the x-axis: 2√3. Since the rectangle is symmetric, we can consider only the positive x-value (√3) and multiply the width by 2 at the end.
The height of the rectangle is given by the absolute value of the y-value: 0.

Step 4: Calculate the area of the rectangle.
The area of a rectangle is given by the product of its width and height: area = width * height = 2√3 * 0 = 0.

Therefore, the maximum area of the rectangle inscribed between the x-axis and the graph of f(x) = -x^4 + 3 is 0.