The sum of the sides of a triangle squared equals ten. (a^2+b^2+c^2=10). Using lagrange multipliers find the minimum perimeter of the triangle.

We want to minimize

f(a,b,c)=a+b+c

subject to the constraint:

a^{2}+b^{2}+c^{2}=10

Thus, with g(a,b,c)=a^{2}+b^{2}+c^{2}, we have:

{\nabla}f(a,b,c)={\lambda}{\nabla}g(a,b,c)

i+j+k={\lambda}(2ai+2bj+2ck)

Equating like terms leads us to:

1=2a{\lambda}

1=2b{\lambda}

1=2c{\lambda}

So, \frac{1}{2a}={\lambda}

\frac{1}{2b}={\lambda}

\frac{1}{2c}={\lambda}

\frac{1}{2a}=\frac{1}{2b}=\frac{1}{2c}

a=b=c

sub this into the constraint and we get:

3a^{2}=10

a=b=c=\sqrt{\frac{10}{3}}

The minimum perimeter is

3\cdot \sqrt{\frac{10}{3}}=\sqrt{30}\approx 5.48