Same question 1 appears again...

V = 2n²x - 6nx² + 4x³

dV/dx = 2n² - 12nx + 12x²

For stationary value of V, dV/dx = 0
2n² - 12nx + 12x² = 0
6x² - 6nx + n² = 0

Using quadratic formula,
x = ( -b ± √(b² - 4ac) ) / 2a
x = ( -(-6n) ± √((-6n)² - 4(6)(n²)) ) / 2(6)
x = ( 6n ± √(36n² - 24n²) ) / 12
x = ( 6n ± √12n² ) / 12
x = ( 6n ± √(3 x 4n²) ) / 12
x = (6n ± 2n√3) /12

(we can square root 4n² to get 2n.
No need to worry about ± sign here as n is positive, so 4n² is also positive and the square root of a positive is represented by the principal root.

eg. √4 = 2 and not -√2 )

x = ½n ± ¹⁄6 n√3

Now,

When x = ½n + ¹⁄6 n√3,

2x = n ± ⅓ n√3

Since n > 0, then ⅓ n√3 > 0

So n + ⅓ n√3 > n and n - ⅓ n√3 < n.

n + ⅓ n√3 > n implies that 2x > n.

That is not possible as the the width of the box must be greater than the sum of the 2 sides of the squares. I.e n > 2x

So x = ½n + ¹⁄6 n√3 is rejected.
Therefore x = ½n - ¹⁄6 n√3

dV/dx = 2n² - 12nx + 12x²

So d²V/dx² = 24x - 12n
= 24(½n - ¹⁄6 n√3) - 12n
= 12n - 4n√3 - 12n
= -4n√3 < 0

(since n > 0, which means 4n√3 > 0)

So the minimum value of V is obtained when x = ½n - ¹⁄6 n√3

Indeed, we don't need a calculator at all to answer this question completely.

https://ask.manytutors.com/questions/67598 if you would like to see the previous version of his working

Thank you

Thank you!!