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MT2_06S w sol

# MT2_06S w sol - YOUR NAME Alexander Givental Math 53...

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YOUR NAME: Alexander Givental Math 53. Midterm II. October 20, 2006. Theoretical question (15 pts) : Constrained extrema and Lagrange’s method. Let S be a surface in R 3 given by the equation g ( x, y, z ) = 0 where g is a diFerentiable function, and let f be another diFerentiable function in R 3 whose domain contains S . Let f | S denote a function S R obtained by restricting the domain of the function f to the surface S . By de±nition of critical points, a point p S is critical for the function f | S , if the directional derivative of f at p in the direction of any vector v tangent to S vanishes: D v f = 0 for all v tangent to S at p . Such critical points are often called constrained extrema of the function f under the constraint g = 0 . Theorem (Lagrange’s method). Constrained extrema of a function f under a constraint g = 0 are in one-to-one correspondence with critical points of the auxiliary function F ( x, y, z, λ ) := f ( x, y, z ) - λg ( x, y, z ) . Proof. At a critical point p = ( x, y, z ) of f | S , for all vectors v tangent to S at p , we have: 0 = D v f = p f · v . Therefore the vector p f is perpendicular to the tangent plane to S at p and is therefore proportional to the vector

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MT2_06S w sol - YOUR NAME Alexander Givental Math 53...

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