1. Inequality and Reasoning problem

Using calculus and reasoning prove how, step by step, the following is true.

a, b, and c are >= 0
If a^2 + b^2 + c^2 + abc = 4 show that ab + bc + ac - abc <= 2

This is a calc 3 problem, vectors and planes in spaces, so i'm thinking i have to find the plane for the first equation and then show relationships to the other inequality. Basically, I'm pretty much lost and have no clue where to start. Any help is appreciated. Thanks.

2. It is more fun to try to prove this inequality without calculus but if you want to use calculus I would start with Lagrange Multipliers.

3. so what would i use as f and g? so would I the gradient of the first equations and set it equal to the gradient of the second, times lamba, and then what? I know how to use lagrange multipliers, but how do you use them in order to prove an inequality? I also made a mistake originally, the equation is suppose to be: a^2 + b^2 + c^2 +abc = 4

4. Originally Posted by jjm5119
Using calculus and reasoning prove how, step by step, the following is true.

a, b, and c are >= 0
If a^2 + b^2 + c^2 + abc = 4 show that ab + bc + ac - abc <= 2
.
So let the constraint function be $g(a,b) = a^2+b^2+c^2 + abc$ and the constraint is $g(a,b) = 4$. Let $f(a,b) = ab+bc+ac-abc$ be this function you are trying to max/minimize. Then that means $\nabla f(a,b) = k\cdot \nabla g(a,b)$. You following? Now try to show using this approach that the maximum attainable value of $f(a,b)$ is $2$ which will complete the demonstration.

5. so the equations i have are:

b+c-bc =k(2a+bc)
a+c-ac =k(2b+ac)
b+a-ab =k(2c+ab)
a^2+b^2+c^2+abc=4

How do you suggest that I start this lagrange? Solve for k?

6. Hey ThePerfectHacker how would i solve this in other ways liike you said? I am stumped on this problem using lagrange mutipliers and would be interested in learning how you would prove this inequality without using them.

Thanks.