Exercise 1: For the following properties, give an example of a functionf that is continuous everywhere and satisfies the given conditions |(a) f(x) f(x) for all xE R (b) f'(1)2, f(1) 7 1 for x < 0, but f'(x)< -1 for (c) f'(x) > 0 (d) f is differentiable everywhere with f"(x) > 0 for all r E R, but f"(x) is not differentiable at x = 0

Q: et [a, b] be a real interva b]→X is continuous, the is continuously differentia

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Q: (0,1). Prove the following functions is uniformly continuous on sin x f(x) =

A: Uniformly Continuous:-  Here we prove the result using a theorem of uniformly continuous.

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Q: กำหนดให้ f(x, y) = xy cos.x x² +3y²

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Q: 2. Determine the points where f is continuous where x + Y if (x, y) # (0,0) f(x, y) = x2 + y2 if (x,…

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Q: I. Show that the function f defined by 1, (x, y) = (1, –1) f(x, y) = x2 + y (x, y) # (1, –1) x +y is…

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Q: Let f(x) be a twice differentiable function on (-o, 0) such that f"(x) is continuous on (-0, 00).…

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Q: Σ sin k+1

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Q: 12. Consider x²y x4 + y? f(x, y) : for (x, y) # (0, 0) and zero otherwise. Is f continuous at (x, y)…

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Q: Let Assume that f has a continuous derivative on a, b. Show tha p(x) such that |f(r) – p(x)| 0. (b)…

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Q: Exercise 1. Express the limit as a definite integral on the given interval cosxi a) lim E-1 Xị Ax,…

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Q: (5) Suppose that fis continuous on [a, b]- f(x) ≥ 0 Vx = [a,b] and [f(x)dx = 0. Prove that f(x) = 0…

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