3. Since𝑆is open,𝑇=𝑓−1(𝑆) is open. Therefore,𝑇=𝑓(𝑆) is a neighborhood of𝑓(x0). Therefore,𝑓is locally onto.4.62Assume to the contrary that there existsx0∕=x1∈𝑆with𝑓(x0) =𝑓(x1). Letx=x1−x0. Define𝑔: [0,1]→𝑆by𝑔(𝑡) = (1−𝑡)x0+𝑡x1=x0+𝑡x. Then𝑔(0) =x0𝑔(1) =x1𝑔′(𝑡) =xDefineℎ(𝑡) =x𝑇(𝑓(𝑔(𝑡))−𝑓(x0))Thenℎ(0) = 0 =ℎ(1)By the mean value theorem (Mean value theorem), there exists 0< 𝛼 <1 such that𝑔(𝛼)∈𝑆andℎ′(𝛼) =x𝑇𝐷𝑓[𝑔(𝛼)]x=x𝑇𝐽𝑓(𝑔(𝛼))x= 0which contradicts the definiteness of𝐽𝑓.4.63Substituting the linear functions in (4.35) and (4.35), the IS-LM model can beexpressed as(1−𝐶𝑦)𝑦−𝐼𝑟𝑟=𝐶0+𝐼0+𝐺−𝐶𝑦𝑇𝐿𝑦𝑦+𝐿𝑟𝑟=𝑀/𝑃which can be rewritten in matrix form as(1−𝐶𝑦𝐼𝑟𝐿𝑦𝐿𝑟) (𝑦𝑟)=(𝑍−𝐶𝑦𝑇𝑀/𝑃)where𝑍=𝐶0+𝐼0+𝐺. Provided the system is nonsingular, that is𝐷=1−𝐶𝑦𝐼𝑟𝐿𝑦𝐿𝑟∕= 0the system can be solved using Cramer’s rule (Exercise 3.103) to yield
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