Let f be a twice-differentiable real-valued function satisfying f(x) + f ″(x) = −xg(x)f′(x), where g(x) ≥ 0 for all real x. Prove that ∣ f(x)∣ is bounded.

Q: snip

A: The given statement is true. Because the local maximum is the highest point on the curve. so that…

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A: NOTE: Refresh your page if you can't see any equations. . find the left-hand limit of the function…

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A: Topic:- Functions

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A: our objective is to find the derivative

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Q: 1. Determine whether k(x) is continuous at a = 3, where x + 3, x 2.

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A: Given,

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A: Topic = Functions

Q: Find c such that the function ² - 4 x c is continuous everywhere.

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Q: Let f(x) x² and g(x) = xf(x). Find c = (0, 1) as guaranteed by the Cauchy Mean Value Theorem.

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Q: A continuous real-valued function f (x) with f (a) ƒ (b) < 0 , has at least one root in [a, b]

A: This question is related to Real Analysis.

Q: Find the value of c such that the function is continuous on the entire real number line.

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A: ANSWER : c = 5

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.