Lagrange Multipliers — Question 1

Question 1 – Solution

Strategy. Solve $\nabla f=\lambda\nabla g$ together with the circular constraint, then compare the finite candidate set.

Step 1: Lagrange system Let $g=x^2+y^2$. Then $$\left\langle 1,1\right\rangle=\lambda\left\langle 2x,2y\right\rangle, \qquad x^2+y^2=1.$$ The first two equations imply $x=y$. Hence $2x^2=1$, so $$(x,y)=\left(\frac 1{\sqrt 2},\frac 1{\sqrt 2}\right) \quad\text{or}\quad \left(-\frac 1{\sqrt 2},-\frac 1{\sqrt 2}\right).$$

Step 2: Values Direct evaluation gives $$\boxed{f_{\max}=\sqrt 2 \text{ at }\left(\frac 1{\sqrt 2},\frac 1{\sqrt 2}\right)},$$ $$\boxed{f_{\min}=-\sqrt 2 \text{ at }\left(-\frac 1{\sqrt 2},-\frac 1{\sqrt 2}\right)}.$$

Step 3: Geometry and verification The level curves $x+y=c$ are parallel lines with normal $\nabla f$. At an extreme one such line is tangent to the circle, so its normal is parallel to the radial normal $\nabla g$. The circle is compact and the two candidates exhaust the multiplier system, completing the verification.