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Where f(n)(a) is the nth derivative of f(x) at x= a Example Find the Taylor series for f(x) = e3x centered. Synthetic Trick Shift ATF 100% w B h b O V ɂ I g } ~ b V t h B @ ʎg p ɂ Ă \ Q ł B N H ^ p b N ݂̂̔̔ ƂȂ ܂ B @ L ̉ i ́A12 { ̉ i ł B P/N. One billion more people benefiting from One billion more people better protected from One billion more people enjoying better U N I V E R S A L H EALTH.

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¶ » n N _ R = 3 Z ˆ Ñ A Ü X ¥ Ð · ° ¸ û h f c Ù. Course Title JASDHA DAMBDA;. Theorem 1 A function f Rn!Ris convex if and only if the function g R!Rgiven by g(t) = f(x ty) is convex (as a univariate function) for all xin domain of f and all y2Rn (The domain of ghere is all tfor which x tyis in the domain of f) Proof This is straightforward from the de nition.

Continuing in this fashion, we nd that in general c n = f(n)(a) n!. A basic theorem of limits says (lim x>a A(x)) (lim x>a B(x)) = lim x>a (A(x) B(x)) when the limits on the left exist It also asserts the existence of the limit on the right. { f B s A X 14G (16mm) O X g g { o x C _ X g A T W J i f B X p { f B s A X j i f B X A N Z T j V i g p y T C Y z14G(16mm) y z1 D ɂȂ ܂ B y f ށz T W J X e X316L ( ×p) y d l z { ̓l W ł B y g p ʁz R N A C i R N A փ b N X A g K X A A ` g K X A _ C X A I s ^ y z 摜 ̓C W ł 肢 ܂ B @ j ^ ̊ ɂ ۂ̃J Ƃ͎኱ ق ꍇ ܂ B @ ʐ^ ɎB e ҂ ʂ肱 ł ꍇ ܂ B y Ӂz.

Theorem 1 A function f Rn!Ris convex if and only if the function g R!Rgiven by g(t) = f(x ty) is convex (as a univariate function) for all xin domain of f and all y2Rn (The domain of ghere is all tfor which x tyis in the domain of f) Proof This is straightforward from the de nition. Ü ˘ ' 3 Ý Þ ˛ ‰ ¾ ^ _ – 2 V ß 6 _ ‰ ž Z $ M ˜ ¢ _ à & À ¸ ˘ ˙ $ _ ž V G ˆ & Ł Ł ˛ _ fi Ö ˚ 8 ˆ ' ¿ fi Z g ˛ d ” b _ l 3 ‰ ¾ ^ _ 3 4 ” ˛ X _ ˘ ˛ ¢ l / $ _ ¾ Ü ™ ˛ ` r ˚ € R À ´ ˛ {Ü _ À V Ô † – ƒ G a 2 _ d h ¯ r ˚ € T ˇ “ µ ˛ 5 ˘ ¡ ˇ ˛ _ Ô ° ¥ ƒ ˆ ˆ c ˘ ˛ r. F is twice differentiable if domf is open and the Hessian ∇2f(x) ∈ Sn, ∇2f(x)ij = ∂2f(x) ∂xi∂xj, i,j = 1,,n, exists at each x ∈ domf 2ndorder conditions for twice differentiable f with convex domain • f is convex if and only if ∇2f(x) 0 for all x ∈ domf • if ∇2f(x) ≻ 0 for all x ∈ domf, then f is strictly convex.

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