# Calculus 3: Lagrange Multipliers

Find the minimum and maximum of $$f(x,y,z) = x^2+y^2+z^2$$ subject to two constraints, $$x+2y+z=8$$ and $$x−y=5$$.

Looking at the equation, it's clear that there is no maximum.

After working this problem out, I found: $$x = 41/11$$ , $$y = -14/11$$ , and $$z = 9/11$$

After plugging this into the original equation, I found the minimum to be $$178/11$$

However, my online homework is saying my answer is incorrect. Did I do something wrong?

Thank you in advance to anyone who can help me out with this.

• To check whether or not you did something wrong, one would need to know what exactly you did. Nov 13, 2018 at 5:11
• The two constraints define a line. What are its min/max distances from the origin? In other words, check your work by solving the problem a different way.
– amd
Nov 13, 2018 at 6:26

• $$y=x-5$$
• $$z=8-x-2y=18-3x$$
$$g(x)=f(x,x-5,18-3x)=x^2+(x-5)^2+(18-3x)^2=11x^2-118x+349$$
I solved equation, and got an answer of $$x=\dfrac{23}{3}$$ ,$$y=\dfrac{8}{3}$$,$$z=-5$$. Is it the correct answer, by your online homework? If it is, I will share my solution with you. By the way, your answer doesn't fit the first constraint, you can check it by plugging in values.