For a closed rectangular box, with a square base $x$ by $x \mathrm{~cm}$ and a height $h \mathrm{~cm}$, find the dimensions giving the minimum surface area, given that the volume is $14 \mathrm{~cm}^{3}$. NOTE: Enter the exact answers, or round to three decimal nlaces

Step 1 ：Given a closed rectangular box with a square base of dimensions \(x \times x \mathrm{~cm}\) and a height \(h \mathrm{~cm}\), we need to find the dimensions that minimize the surface area, given that the volume is \(14 \mathrm{~cm}^{3}\).

Step 2 ：The surface area (SA) of the box is given by the formula: \(\mathrm{SA} = 2(x^2) + 4(xh)\).

Step 3 ：The volume (V) of the box is given by the formula: \(\mathrm{V} = x^2h\).

Step 4 ：We are given that the volume is \(14 \mathrm{~cm}^{3}\), so we have: \(x^2h = 14\).

Step 5 ：Solve for h in terms of x: \(h = \frac{14}{x^2}\).

Step 6 ：Substitute this expression for h back into the surface area formula: \(\mathrm{SA} = 2(x^2) + 4\left(x\left(\frac{14}{x^2}\right)\right)\).

Step 7 ：Take the derivative of the surface area function with respect to x and set it equal to 0: \(\frac{d\mathrm{SA}}{dx} = 4x - \frac{56}{x^2} = 0\).

Step 8 ：Solve for x: \(x = \sqrt[3]{14}\).

Step 9 ：Find the corresponding value for h using the expression \(h = \frac{14}{x^2}\): \(h = \sqrt[3]{14}\).

Step 10 ：\(\boxed{\text{Final Answer: The dimensions of the closed rectangular box with a square base and minimum surface area, given that the volume is 14 cm³, are approximately x = 1.913 cm and h = 1.913 cm.}}\)