Centroid = ((0+6+0)/3, (0+0+8)/3) = (2, 2).

Midpoint M = ((2+4)/2, (-1+3)/2, (3+5)/2) = (3, 1, 4).

Midpoint M = ((2+4)/2, (3+5)/2, (4+6)/2) = (3, 4, 5).

f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 0 is a critical point.

Setting f'(x) = 4x^3 - 16x = 0 gives x = 0, ±2. Evaluating f(2) = 0 shows (2, 0) is a critical point.

The derivative f'(x) = d/dx(sin(x) + cos(x)) = cos(x) - sin(x).

The derivative f'(x) = d/dx(tan(x)) = sec^2(x).

The derivative f'(x) = d/dx(x^5 + 3x^3 - 2x) = 5x^4 + 9x^2 - 2.

The derivative f'(x) = d/dx(x^5) - d/dx(3x) + d/dx(2) = 5x^4 - 3.

The derivative f'(x) = d/dx(x^5 - 3x^3 + 2) = 5x^4 - 9x^2.

Determinant of E = 0 (rows are linearly dependent).

Det(E) = (3*4) - (2*1) = 12 - 2 = 10.

Det(E) = (4*3) - (2*1) = 12 - 2 = 10.

Det(F) = (4*1) - (3*2) = 4 - 6 = -2.

Det(F) = (4*7) - (5*6) = 28 - 30 = -2.

Determinant of G = (1*4) - (2*2) = 4 - 4 = 0.

Determinant of H = (3*5) - (1*2) = 15 - 2 = 13.

The determinant of J is calculated as (5*3) - (2*1) = 15 - 2 = 13.

The determinant of D can be calculated using the rule of Sarrus or cofactor expansion, which results in 0.

The determinant of D is calculated as (4*1) - (2*3) = 4 - 6 = -2.

The determinant is \( 3*4 - 2*1 = 12 - 2 = 10 \).

The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).

The determinant is calculated as \( 2*7 - 3*5 = 14 - 15 = -1 \).

The determinant is \( 4*1 - 2*3 = 4 - 6 = -2 \).

Let x be the side of the base and h the height. The surface area constraint gives 2x^2 + 4xh = 600. Max volume occurs at x = 12.

For a fixed area, the minimum perimeter occurs when the rectangle is a square. Thus, dimensions are approximately 7 m by 7.14 m.

For minimum perimeter, the rectangle should be a square. Thus, side = sqrt(50) ≈ 7.07.

For a fixed area, the perimeter is minimized when the rectangle is a square. Thus, side = √50.

Distance = |d1 - d2| / √(a² + b² + c²) = |5 - 10| / √(2² + 3² + 4²) = 5 / √29.

Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + 1²) = 5 / √14.