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Chemical Engineering Hand Written_Notes_Part_82

Chemical Engineering Hand Written_Notes_Part_82 - 166 5...

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166 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Theorem 11 . If F ( z ) is continuous and di ff erentiable and has an extreme (or stationary) point (i.e. maximum or minimum ) point at z = z , then (2.1) F ( z ) = ∂F ∂z 1 ∂F ∂z 2 .............. ∂F ∂z N ¸ T z = z = 0 . Proof: Suppose z = z is a minimum point and one of the partial deriva- tives, say the k th one, does not vanish at z = z , then by Taylor’s theorem (2.2) F ( z + z ) = F ( z ) + N X i =1 ∂F ∂z i ( z ) z i + R 2 ( z , z ) (2.3) i.e. F ( z + z ) F ( z ) = z k ∂F ∂z i ( z ) + R 2 ( z , z ) Since R 2 ( z , z ) is of order ( z i ) 2 , the terms of order z i will dominate over the higher order terms for su ciently small z . Thus, sign of F ( z + z ) F ( z ) is decided by sign of z k ∂F ∂z k ( z ) Suppose, (2.4) ∂F ∂z k ( z ) > 0 then, choosing z k < 0 implies (2.5) F ( z + z ) F ( z ) < 0 F ( z + z ) < F ( z ) and F ( z ) can be further reduced by reducing z k . This contradicts the assump- tion that z = z is a minimum point. Similarly, if
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