Question

Maximum Volume The volume of an ellipsoid \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1\) is \(4 \pi a b c / 3\). For a fixed sum \(a+b+c\), show that the ellipsoid of maximum volume is a sphere.

Short answer

When \(a = b = c = \frac{k}{3}\), the ellipsoid becomes a sphere with maximum volume under the given condition, confirming the claim to prove.

Step-by-step solution

Define the Function and the Constraint

The volume function is \(V(a, b ,c) = \frac{4}{3}\pi abc\). The constraint function is \(g(a,b,c) = a + b + c - k = 0\), where \(k\) is the fixed sum.

Formulate the Lagrange Function

The Lagrange function \(L(a, b ,c , \lambda) = V(a, b ,c) - \lambda g(a,b,c)\) combines the objective function and the constraint by subtracting the product of a Lagrange multiplier \(\lambda\) and the constraint function from the objective function. Thus noted as: \(L(a, b ,c , \lambda) = \frac{4}{3}\pi abc - \lambda(a + b + c - k)\)

Compute the Partial Derivatives and form the System of Equations

Find the partial derivatives of the Lagrange function with respect to \(a\), \(b\), \(c\), and \(\lambda\) and set each derivative equal to zero : \[\frac{\partial L}{\partial a} = \frac{4}{3}\pi bc - \lambda = 0\] \[\frac{\partial L}{\partial b} = \frac{4}{3}\pi ac - \lambda = 0\] \[\frac{\partial L}{\partial c} = \frac{4}{3}\pi ab - \lambda = 0\] \[\frac{\partial L}{\partial \lambda} = k - a - b - c = 0\]

Solve the System of Equations

Solving the system of equations gives the critical points which potentially could maximize or minimize our function under the given constraint. From the first three equations we have \(\frac{4}{3}\pi bc = \frac{4}{3}\pi ac = \frac{4}{3}\pi ab = \lambda\), implying that \(a = b = c\) and along with the fourth equation, we get \(a = b = c = \frac{k}{3}\)

Verifying Maximization Condition

The values found give a local maximum to the volume function. It can be verified using second derivative test or observing that this is the only critical point we got. Also any ('a', 'b', 'c') other than ('k/3', 'k/3', 'k/3') will have at least one of 'a', 'b', 'c' to be less than 'k/3', leading to smaller volume.