Generalizing proofs is a crucial mathematical skill. Here you will prove the method of Lagrange multipliers with multiple constraints by generalizing the textbook's proof for a single constraint. Let URn be open. Assume f:UR is differentiable, g1,,gk:UR are C1, and c1,,ckR. Let S={xU:g1(x)=c1,,gk(x)=ck} Assume S is non-empty and for every qS that the set {g1(q),,gk(q)} is linearly independent. (4a) Use Theorem 5.5.7 to prove that S is a (nk)-dimensional smooth manifold and, for every pS, the tangent space TpS is a (nk)-dimensional subspace of Rn given by TpS={vRn:i{1,,k},gi(p)v=0} W={wRn:vTpS,vw=0}. Here is a linear algebra result which you may use without proof. Lemma. If V is an -dimensional subspace of Rn, then the set {wRn:vV,vw=0} is a (n)-dimensional subspace of Rn. Use this lemma and (4a) to conclude that W=span{g1(p),,gk(p)}. c) Prove that if p is a local extremum of f on S, then there exists 1,,kR such that f(p)=j=1kjgj(p). This completes the proof of Lagrange multipliers for multiple constraints! Generalizing proofs is a crucial mathematical skill. Here you will prove the method of Lagrange multipliers with multiple constraints by generalizing the textbook's proof for a single constraint. Let URn be open. Assume f:UR is differentiable, g1,,gk:UR are C1, and c1,,ckR. Let S={xU:g1(x)=c1,,gk(x)=ck} Assume S is non-empty and for every qS that the set {g1(q),,gk(q)} is linearly independent. (4a) Use Theorem 5.5.7 to prove that S is a (nk)-dimensional smooth manifold and, for every pS, the tangent space TpS is a (nk)-dimensional subspace of Rn given by TpS={vRn:i{1,,k},gi(p)v=0} W={wRn:vTpS,vw=0}. Here is a linear algebra result which you may use without proof. Lemma. If V is an -dimensional subspace of Rn, then the set {wRn:vV,vw=0} is a (n)-dimensional subspace of Rn. Use this lemma and (4a) to conclude that W=span{g1(p),,gk(p)}. c) Prove that if p is a local extremum of f on S, then there exists 1,,kR such that f(p)=j=1kjgj(p). This completes the proof of Lagrange multipliers for multiple constraints