what I need for is #2!
#1 is attached for #2.
Please help me! Thanks
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what I need for is #2! #1 is attached for #2. Please help me! Thanks 1. In class we showed that the function f : R → R...
plz help me !! Thanks 1. In class we showed that the function f : R → R given by (if>o 0 if a S0 was smooth (but not real analytic). Note that f(x) approaches a horizontal asymptote (y = 1) as a goes to positive infinity. (a) Show that f(x)+f(1-2)メ0 for all x E R, so that g : R → R given by g(x)- 70 is also a smooth function. (b) Prove that if 0 ifx-1. (c) Note...
plz help me analysis question! Thanks in advance 2. Let h : R-+ R be the smooth function given by h(z) g is as in Problem 1 g(z + 2g(2-x) for all r E R, where (a) Show that if a < -2 0 g(2) if -2< <-1 h(x) if 2 0 (b) Use part (d) of Proble 1 to show that for all E 0,9 in fact for all ,. Conclude that for all e 0,1 The functions from...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Please answer this question Implicit Function Theorem in Two Variables: Let g: R2 - R be a smooth function. Set Suppose g(a, b)-0 so that (a, b) є S and dg(a, b) 0. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above (2) Since dg(a, b)メ0, argue that it suffices to assume a,b)メ0. (3) Prove the...
need help with all a, b, c 2. 15 Marks (a) Suppose that f : R" R is convex but not necessarily smooth. Prove that h-af is a (b) Suppose that f : R -R is convex and smooth. Also assume that f(x) > 0 for all z (c) Show that the set S = {(x,y) : y > 0} is convex and that the function f(x,y)-x2/v is convex function if a-0. Show with a simple example that this is...
2) (12) f:R-(3/2)-R-10, (x) 1/(3 2x) g:R--21->R-1o), g (x)1/ (x 2) h:R-(-4/3]-R-(1/3), h(x) (f o g) (x) Verify if h(x) is one to one and onto. If it is, find the inverse function of h(x). 2) (12) f:R-(3/2)-R-10, (x) 1/(3 2x) g:R--21->R-1o), g (x)1/ (x 2) h:R-(-4/3]-R-(1/3), h(x) (f o g) (x) Verify if h(x) is one to one and onto. If it is, find the inverse function of h(x).
Let f: C→C be an entire, one-to-one function. (a) Explain why g()-f() f(0) is an entire 1-1 function (b) Explain why there exists0 such that B(O,e) C g(B(O, 1)). Hint: Open Mapping thm.] (c) Explain why Ig(z)2є if 221 . [Hint: g is 1-1.] (d) Since g(0)=0, g(z)=2h(z) for some entire function h(z). Explain why h(z) is never 0 (e) Show that there is a constant C>0 such that 1/h2)l C if21 (f) Deduce that 1/h (z) is a constant...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
number 1 and 2 pls Problem 1.1. Suppose that f: R → R and that f is differentiable at z = a. 1. Show that, given an angle 6, we can choose 6(0) > 0 small enough so that for all r such that r - al < (0) we have that the graph of f(r) lies inside of the cone with angle e around the tangent line. 2. Can you find explicit formulas for 6(0) for the function f(x)...