Question

Use Lagrange multipliers to prove that the triangle with maximum area that has a given perimeter\(p\)is equilateral. (Hint: Use Heron's formula for the area: \(A = \sqrt {s(s - x)(s - y)(s - z)} \), where\(s = p/2\)and\(x,y,z\)are the lengths of the sides.)

Short answer

So, the triangle with maximum area that has given perimeter\(p\)is an equilateral triangle.

Step-by-step solution

Method of Lagrange multipliers 

To find the maximum and minimum values of\(f(x,y,z)\)subject to the constraint\(g(x,y,z) = k\)(assuming that these extreme values exist and\(\nabla g \ne {\bf{0}}\)on the surface\(g(x,y,z) = k)\):

(a) Find all values of\(x,y,z\), and\(\lambda \)such that\(\nabla f(x,y,z) = \lambda \nabla g(x,y,z)g(x,y,z) = k\)

And

(b) Evaluate\(f\)at all the points\((x,y,z)\)that result from step (a). The largest of these values is the maximum value of\(f\); the smallest is the minimum value of\(f\).

Assumption

Let the sides of the triangle are\(x,y,z\)and\(p\)is the perimeter of triangle.

Then\(A(x,y,z)\), the area of the triangle is\(A(x,y,z) = \sqrt {(s(s - x)(s - y)(s - z))} \)

Where\(s = \frac{{x + y + z}}{2} = \frac{p}{2}\) where \(s\)is also constant.

Let\(f(x,y,z) = s(s - x)(s - y)(s - z)\) and\(g(x,y,z) = x + y + z - p\)

Use Lagrange multiplier

Using Lagrange's method to find all\(x,y,z\)such that\(\vec \nabla f(x,y,z) = \lambda \vec \nabla g(x,y,z)\)

That is\(\vec \nabla f(x,y,z) = \lambda \langle 1,1,1\rangle \)since \(g(x,y,z) = x + y + z - p\)

\(\begin{array}{l}s( - 1)(s - y)(s - z) = \lambda \ldots \ldots (1)\\s(s - x)( - 1)(s - z) = \lambda \ldots \ldots (2)\\s(s - x)(s - y)( - 1) = \lambda \ldots \ldots (3)\\x + y + z = p\end{array}\)

Solve equation

By equation (1), (2):

\(\begin{array}{c}s( - 1)(s - y)(s - z) = s(s - x)( - 1)(s - z)\\(s - y) = (s - x)\\y = x\end{array}\)

From equation (2), (3):

\(\begin{array}{c}s(s - x)( - 1)(s - z) = s(s - x)(s - y)( - 1)\\(s - z) = (s - y)\\z = y\end{array}\)

So\(x = y = z\).

Find length of side

Substitute in\(x + y + z = p\)

\(\begin{array}{l}3x = p\\x = \frac{p}{3}\end{array}\)

So, \(x = y = z = \frac{p}{3}\)

Therefore lengths of triangle are equal.

Hence the triangle with maximum area that has given perimeter\(p\)is an equilateral triangle.