The perimeter of a rectangle is $24 \mathrm{~cm}$. Find the lengths of the sides of the rectangle giving the maximum area.

Step 1 ：Let the side lengths of the rectangle be x and y. The perimeter is given by the formula P = 2x + 2y, and the area is given by the formula A = xy.

Step 2 ：Since the perimeter is 24 cm, we have the equation: \(2x + 2y = 24\)

Step 3 ：Solve for y: \(y = 12 - x\)

Step 4 ：Substitute the expression for y into the area formula: \(A = x(12 - x)\)

Step 5 ：Find the critical points by taking the derivative of the area function and setting it to 0: \(\frac{dA}{dx} = 12 - 2x = 0\)

Step 6 ：Solve for x: \(x = 6\)

Step 7 ：Substitute x back into the expression for y: \(y = 12 - 6 = 6\)

Step 8 ：\(\boxed{\text{Final Answer: The side lengths of the rectangle giving the maximum area are 6 cm and 6 cm}}\)