Solve 1/a + 1/b + 1/c When a+b+c+2=abc

[Hobold]

A friend of mine gave me this problem to solve because he couldn't and I've been stuck in it for some time and also can't solve.

If a, b, c are positive real numbers and a + b + c + 2 = abc, find the smallest value of 1/a + 1/b + 1/c.

PS: I'm guessing this is the right section, 'cause this is not a homework.

 

[physi-man]

The only thing I can see that fits for the first part is a, b, and c are all 2.
2+2+2+2=8 and 2x2x2=8. So the last part would be 1.5
smallest value of 1/2 + 1/2 + 1/2=1.5
Unless I am reading something wrong.

 

[Hobold]

The only thing I can see that fits for the first part is a, b, and c are all 2.
2+2+2+2=8 and 2x2x2=8. So the last part would be 1.5
smallest value of 1/2 + 1/2 + 1/2=1.5
Unless I am reading something wrong.

Yep, I got this result as well, but there's nothing that proves that there isn't a combination of numbers which will make 1/a + 1/b + 1/c < 1.5

 

[Office_Shredder]

There are two ways that you can do this:
1) Use Lagrange multipliers. You're trying to minimize a function given a constraint so it would work, but might be ugly.

2) Square something crazy. This looks like it's probably a high school math competition problem, which means that 1 is an inappropriate solution. Instead you probably need to do something like square something, which you know is greater than or equal to zero, or maybe use AM/GM. I don't see exactly what it is you would be using but I'll think about it

As for other solutions:

a(1-bc)+b+c+2=0

(-2-b-c)/(1-bc)=a. Now pick values of b and c making this positive. For example if b=c=3, we get (-2-6)/(1-9)=-8/-8=1. So (1,3,3) is also a solution

Of course this isn't a better solution, but you never know if there is one. The symmetric solution is usually optimal when the problem is symmetric.

 

[CellCoree]

I would approach this problem by first setting up the equation and then manipulating it to see if any patterns or relationships emerge. In this case, we have the equation a + b + c + 2 = abc, and we want to find the smallest value of 1/a + 1/b + 1/c.

One way to approach this problem is to use the method of substitution. We can rearrange the original equation to get c = (a + b + 2)/(ab - 1). We can then substitute this value of c into the expression 1/a + 1/b + 1/c to get 1/a + 1/b + (ab - 1)/(a + b + 2). This expression can be simplified to (ab + a + b + 1)/(ab(a + b + 2)). This shows that the expression 1/a + 1/b + 1/c is dependent on the values of a and b.

To find the smallest value of 1/a + 1/b + 1/c, we can use the method of differentiation. Taking the derivative of the expression with respect to a and setting it equal to 0, we get the equation (ab + b + 1 - a^2b)/(ab(a + b + 2)^2) = 0. Solving for a, we get a = (b + 1)/(b - 1). Similarly, taking the derivative with respect to b and setting it equal to 0, we get b = (a + 1)/(a - 1). Substituting these values into the expression 1/a + 1/b + (ab - 1)/(a + b + 2), we get the smallest value as 3/2.

In conclusion, the smallest value of 1/a + 1/b + 1/c is 3/2 when a = (b + 1)/(b - 1) and b = (a + 1)/(a - 1). I would recommend double-checking the steps and values to ensure accuracy. Additionally, this problem may have multiple solutions and it may be helpful to explore other methods of solving it.