if x,y,z > 0 and x+y+z = 3 . Find min T = 1/sqrt x + 1/sqrt y + 1/ sqrt z

Asked on by snowcat97

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nathanshields | High School Teacher | (Level 1) Associate Educator

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Three positive numbers add up to 3.

Let z = 3-x-y

Now let's look at the function we must minimize:

`T(x,y) = 1/sqrt(x)+1/sqrt(y)+1/sqrt(3-x-y)`

A little help from a 3D graph (http://www.wolframalpha.com/input/?i=minimum+T%28x%2Cy%29+%3D+1%2Fsqrt%28x%29%2B1%2Fsqrt%28y%29%2B1%2Fsqrt%283-x-y%29) and we see that the mimum occurs at x = 1, y = 1, (so z = 1)

Finding this solution algebraically requires multivariable calculus, which I figured you haven't studied yet (though I might be wrong).

Another good method would be guessing and checking.  At (1, 1, 1), T = 3.  Any other choice means that at least one of the numbers will be less than 1, and its square root will also be less than 1, so its reciprocal will be greater than 1...

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