Show that if g: R R is a function which is differentiable and with g' bounded (say |g'(x) < M for all x E R), then the map f: R → R defined by f(x) = x+eg(x) for e > 0 is one-to-one for all sufficiently small .

Q: 5. Let f : R → [1, ∞) and g : [1, 0) → R be the functions defined by the following formulæ f (x) =…

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Q: 3.23 Prove or give a counterexample: The function |f| € R[a, b] + ƒ € R{a, b].

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Q: 25. Let f: R→ R and let f(x) > 0 for all x € R. Show that f(x) is strictly decreasing if and only if…

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Q: (b) A continuous function f : [0, 1] → R such that its graph (r, f (x)) C R² is not closed.

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Q: 2. Let f(x) = V4 – x for x < 4 and g(x) = x² for all x E R. (a) Give the domain of f · g, ƒ + g, ƒ º…

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Q: If x + 1 f(x) then find the global minimum of f on the interval [1, c∞)

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Q: Consider the function g defined by x x x + 4

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Q: 30 Chapter 4 Differentiation example of a function that fails to satisfy a Lipschitz condition at a…

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Q: (5) Let f: R→R, and suppose there exist a positive integer M such that 0 < f{x) < M for all x ER.…

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Q: Let ƒ : R → R be a function that is differentiable on (a, ∞), where a is any real onstant. Let g: RR…

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Q: 7. We define f: R R as follows: {. f(x) = X -x3 if x ≥ 1, if 0 < x < 1, if x < 0. (a) Calculate…

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Q: 1. Show that if f: [a,b] → R and a < xo <b, then f'(xo) exists if and only if D f(x) = Dƒ(x) € R.

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