Nope, not $x=y=z$

Algebra Level 5

Let $a, b, c$ be positive reals so that $3ab+bc+2ca=6$ . Find the maximum value of

$\frac { 1 }{ { a }^{ 2 }+1 } +\frac { 4 }{ { b }^{ 2 }+4 } +\frac { 9 }{ { c }^{ 2 }+9 }$

Write your answer to 4 decimal places.

The answer is 2.25.

1 solution

Zach Bian
Dec 29, 2017

I think I accidentally solved it. I got 2.236545652907476.

// Calculate square to save typing and eyestrain.
var sq = (x) => (x*x);

// Our function
var mystery = (a,b,c) => (1/(a*a+1) + 4/(b*b+4) + 9/(c*c+9));

// c solved for a and b
var c = (a,b) => ((6-3*a*b)/(2*a+b));

// The directional derivative of the mystery function. c is subbed in using the above function.
var grad = (a,b) => {
    let clocal = c(a,b);
    return [-2*a/sq(a*a+1),
    -8*b/sq(b*b+4),
    -18*clocal/sq(clocal*clocal+9)
]};

// Magnitude of a vector
var magnitude = (vec) => {
    let acc = 0;
    for(let el of vec){
        acc += el*el;
    }
    return Math.sqrt(acc);
};

// Normalize vector to a size.
var normalize = (vec, factor) => {
    let mag = magnitude(vec);
    if(factor === undefined)
        factor = 1;
    return [...vec.map(a=>(a/mag*factor))];
};

// Heuristic step size:
var stepsize = .0001;

// Start position:
var position = [1,1,c(1,1)];

// Timeout:
var steps = 1000;

// Keep track of maximum:
var max = mystery(...position);

// Climb the hill:
while(steps > 0){
    // Determine direction:
    let stepVec = normalize(grad(position[0], position[1]), stepsize);

    // If step size is too small, exit.
    if(magnitude(stepVec) < 1e-6)
        break;

    // If we go negative, exit.
    if(position[0] + stepVec[0] < 0 || position[1] + stepVec[1] < 0)
        break;

    // Take the step
    position[0] += stepVec[0];
    position[1] += stepVec[1];
    position[2] = c(position[0], position[1]);

    let cache = mystery(...position); 
    if(max < cache){
        max = cache;
    }

    steps--;
    // Uncomment to view steps:
    // if(steps%10==0)
    //     console.log(stepVec, position, max);
}

console.log(max);

Did I go wrong somewhere?

Nope, not x = y = z x=y=z x = y = z

The answer is 2.25.

1 solution

0 pending reports

Nope, not $x=y=z$