Question

Determine the area of the largest rectangle which can be formed under the curve f(x)=5(1/2)^x in the first quadrant.

Please answer using grade 12 calculus knowledge

Answer 1

Answer:

You will get a rectangle of sides a and b, whose area is A=a⋅b, and perimeter L=2a+2b. One approach is calculus: Let x=a, then b=L2−x and area is A=x(L2−x)=L2x−x2, have the derivative equal 0, and voila.

Answer 2

Answer:

If each horizontal side of the rectangle has length x, then the vertical sides each have length f(x) = -6x2 + 1458.

Let A(x) = area of rectangle

A(x) = x(-6x2 + 1458) = -6x3 + 1458x

A'(x) = -18x2 + 1458, x > 0

Set A'(x) equal to zero to locate the critical points of A(x):

-18x2 + 1458 = 0

x2 = 81 So, x = 9

Check the sign of A'(x) on both sides of the critical point:

If 0 < x < 9, A'(x) > 0. So, A(x) is increasing when 0 < x < 9.

If x > 9, A'(x) < 0. So, A(x) is decreasing when x > 9.

Therefore, A(x) has a relative maximum (and absolute maximum) when x = 9.

Maximum area = A(9) = -6(9)3 + 1458(9) = 8748.

Step-by-step explanation:

hope this will help you

please mark it as the brainliest