(9) (Schwartz space) The Schwartz space on R, S(R) is the set of all C* functions f such that its derivatives f',f(2),.,f(k),... decreasing in the sense that supeR |æ|*| f(O (x)| < ∞ for all k, l > 0. Show that f(x) = e=** € S(R) but e¯-l#| ¢ S(R). are all rapidly

Q: Find all the local minima of the function f(x, y) = 15 x 2 x +3 y +6xy a) A local minimum occurs at…

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Q: The range of the function f(x, y) = 6 – V9 – x2 - y? is O (-∞, 6] O (-0, 3] O [3, 00) O [6, 00) O…

A: Given:     f(x,y)=6-9-x2-y2    Range of the function is         9-x2-y2≥0          9≥x2+y2…

Q: 2x²y² {** x² + y² 0 (x, y) = (0,0) (x, y) = (0,0) continuity to determine whether the given function…

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Q: The shaded area of figure is best described by U B O (AUB)'u (ANB) O B' O (AUB) N (ANB) O ANB' None…

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Q: Theorem: Let f,g € C(X)are continuous function then 1) f + g is continuous function and (f + g)(x) =…

A: To prove the given statements.

Q: p) State Cauchy's Schwarz Inequality and prove this for Space of Continuous functions on[-n, n].

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Q: If f(x) = 1+ then the interval(s) of decreasing is(are) Select one: O (-00,-1]U[1, 0) O -1,0) U (0,…

A: f(x) = x + 1x Differentiating with respect to x: f'(x) = 1 - 1x2   .......................using…

Q: Draw a graph for following f' > 0 on (-∞,1) and f"l > 0 on (-∞, 1) U (1, ∞).

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Q: (C0]) }

A: The statement is false. Consider f:[0,1]→ℝ defined as f(x)=tan-1πx2   x∈[0,1)0              x=1

Q: Consider the function a) b) Prove function & R R given by Prove there that Prove f'(e) f(x) there…

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Q: ,10 If f is a continuous function and F'(x)=f(x) for all x € R, then A 2[F(5) – F(1)] B 2[F(10) –…

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Q: Sketch the graph of f(x) if ...f'(x)&gt;0 on (2,4) and f'(x)&lt;0 on (−∞, 2) ∪ (4, ∞),f''(x)&gt;0 on…

A: Here we suppose that the given function is continuous on the given intervals and differentiable on…

Q: 1.8 Exercise Prove that if (X,X) is a cut, then f(X,X)< c(X,X) or equivalently, IS| < c(X,X)

A: Given that X,X is a cut. The by the definition of cut, X∪X=S,X≠∅,X≠∅,X∪X=∅ And if a∈X,b∈X→a&lt;b And…

Q: 3. Iffis differentiable for all z, then we say that fis an entire function. If f and g are entire…

A: The entire function is defined as the function for every point function that gives the finite value…

Q: Let f: X → Y and g: Y → V be functions. Show the following: (a) f is injective 3h : Y→ X such that…

A: Given Data: Let f:X→Y and g : Y→V be functions.To Show that,(a) f is injective ⇔∃h :Y→X  such that h…

Q: Solve using the method of undetermined coefficients: y" + 3y" – 10y' = 2 + xe* %3D

A: Given: y''' + 3y'' - 10y' = 2 + xex

Q: Let 00 f(x) = -1)" I +n n=0 (a) Show f is defined for a > 0. (b) Show that ƒ is continuous on (0,…

A: Other

Q: Let fn(x) = n²x²(1 – x) on [0, 1]. Does lim,oo fn(x) exist for every r E [0, 1]?

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Q: 3. How is the continuity of the following function at x=0? There exist Left continuity Right…

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Q: 2) Show that the Cantor function is not differentiable for any x in the Cantor set, and that the…

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Q: (b) Find an expression az + ba? + ca +d so that this new polynomial is orthogonal to all the…

A: Solution of part (b): The inner product on continuous functions on -2,2 is defined by…

Q: 11. Prove that if f is continuous on (a, b] and lim,a* f(x) exists, then fis uniformly continuous on…

A: The theorem is done .

Q: Use a software program or a graphing utility with matrix capabilities to write v as a linear…

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Q: Q/ Provethet f is continuous if f(B) = f'(B) Suppose of continuous and Prove F(B) ≤ f(B) then for…

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Q: Let f be continuous over [a,b] and differentiable over (a,b). Then there exists at least one point…

A: Let f be continuous over a,b and differentiable over a,b. Then there exits at leastone point c∈a,b…

Q: 10 If f is a continuous function and F'(x)=f(x) for all x € R, then dx=? @2[F(5) – F(1)] ®2[F(10) –…

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Q: If m, n ∈ N, then (f^m)^n = f^(mn).

A: To prove two functions are equal, we need to show that both the functions have same function values…

Q: f(x,y) F(x,y)= (x,y)+ (0,0) (x,y)= (0,0)

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