# About using lagrange multiplier for the min value of $3x+4y$ when $x^2+y^2\leq 16$ and $x,y \in \mathbb{R}$

About this question I have asked lately $x^2+y^2 \leq 16$ and also $\{x,y\} \subset \mathbb{R}^+$ what is the minimum value of $3x+4y$.

I want to solve it with Lagrange now, but I couldn't manage to. I have tried to construct a multi-variable function which was the following $f(x,y)=3x+4y$ then to find the minimum value of it, but I am really not accustomed to using derivatives for extremum problems. And I couldn't find the connection with it between the Lagrange multiplier. What do you suggest?

Let $K=\{(x,y) \in \mathbb R^2:x^2+y^2 \le 16\}$. We observe that there no points$(x_0,y_0)$ such that $f_x(x_0,y_0)=0$ and $f_y(x_0,y_0)=0$.

Consequence: $\min f(K)= \min f( \partial K)$

Hence you have to minimize the function $f$ under the condition $x^2+y^2=16$.

Can you proceed from here ?

• So what I have to do is $f(x,y,k)=3x+4y+k(x^2+y^2-16)$ ? – Deniz Tuna Yalçın Sep 7 '17 at 9:17
• I couldn't find a decent solution using this: $3+2kx=4+2ky$ and I can't keep going – Deniz Tuna Yalçın Sep 7 '17 at 9:18
• Yes, but since $f(x,y)=3x+4y$ you shoul write $g(x,y,k)=3x+4y+k(x^2+y^2-16)$. – Fred Sep 7 '17 at 9:19
• I forgot that, but that didn't change the deriving process $g_x'=3+2kx$ and $g_y'=4+2ky$ and $g_k'=0$ so $3+2kx=4+2ky$. How should I proceed, or am I making a mistake? – Deniz Tuna Yalçın Sep 7 '17 at 9:21
• You have to solve the system $3+2kx=0, \quad 4+2ky=0$ – Fred Sep 7 '17 at 9:22

By C-S $$3x+4y\geq-|3x+4y|\geq-\sqrt{(3^2+4^2)(x^2+y^2)}\geq-\sqrt{25\cdot16}=-20.$$ The equality occurs for $(3,4)||(x,y)$ and $x^2+y^2=16$, which says that $-20$ is a minimal value.

• In the question text I have written $x,y \in \mathbb{R}^+$ – Deniz Tuna Yalçın Sep 7 '17 at 10:05
• Although this will be helpful to evaluate negative expressions with C-S. Thank you:) – Deniz Tuna Yalçın Sep 7 '17 at 10:06
• @Deniz Tuna Yalçın You are welcome! – Michael Rozenberg Sep 7 '17 at 10:07

With Cauchy-Schwarz:

$|3x+4y| \le |(3,4)*(x,y)| \le ||(3,4)||*||(x,y)|| = 5*4=20$

hence:

$-20 \le 3x+4y$.

Now look for $(x,y)$ such that $-20 = 3x+4y$ and $x^2+y^2=16$.

• I recall writing $x,y \in \mathbb{R}^+$ although I have always known C-S to be a method only for positive numbers. Seeing this usage is helpful, thank you:) – Deniz Tuna Yalçın Sep 7 '17 at 10:08

1) $x^2 + y^2 = 16,$ and interior $(\lt )$,

equation of a circle about $O(0,0)$ , and radius $= 4$.

2) Straight line: $3x + 4y = C$, or

$y = -(3/4)x + C/4$.

Slope $m= -3/4$; $Y -$ intercept: $C/4.$

We are looking for a straight line with minimal $Y-$intercept $C/4$.

Three cases:

A) The line does not intersect the circle. Ruled out.

2) The line intersects the circle twice. We can still move in neg. $Y-$ direction to decrease $C/4$

3) The line touches the circle (tangent line).

There are 2 Points where the line is tangential to the circle:

$P_1$, $P_2$, and

distance $OP_1 = OP_2 = 4$.

Using the distance formula for $3x + 4y - C = 0$:

$4 = \frac{-C}{^+_- \sqrt{3^2+4^2}}$

Hence:

$C_{min} = -20,$ $C_{max} = 20$.