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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?Provide an example or explain why the request is impossible.Let’s take the domain of the functions to be all of R. (c) A sequence (fn) of differentiable functions such that both (fn) and (f'n)converge uniformly but f = lim fn is not differentiable at some point.Consider the sequence of functions {fn}nen, where nx fn(z) = Va € [1, 10]. 1+ nx' (a) Find the pointwise limit of the sequence {f} on the interval [1, 10]. (b) Show that the convergence is uniform. .10 (c) Evaluate lim f. fn(x) dx. 12→∞0
- Find all possible value of x for convergence and divergence bothShow whether or not fn(X) =sin (x/n) is pointwise convergenceLet (fn) be a sequence of functions that each fn is continuous [0, 1] and differentiable on (0, 1). Suppose (fn) converges uniformly to a function f on [0, 1]. Which of the following statement (s) must be true? (i) of = lim, So fn. (ii) f is uniformly continuous on [0, 1] (iii) f is differentiable on (0, 1).
- Is Uniform convergence of {fn(x)} is sufficient but not necessary to transmit continuity from the individual terms to the limit function ?Consider the sequence of functions {fn} defined on the interval (1, 0), where fn (x) 1 Vx E (1, 00), Vn E N. Let f be the pointwise limit of { fn}. Determine f. (1,00)Find the intervals of convergence of f(x), f'(x), f"(x), and f(x) dx. (Enter your answer using interval notation. Be sure to check for convergence at the endpoints of the interval.) Š (-1)" + 1(x – 2)" f(x) in n = 1 (a) f(x) (b) f'(x) (c) f"(x) (d) f(x) dx
- Find the intervals of convergence of f(x), f '(x), f"(x), and f(x) dx. Include a check for convergence at the endpoints of the interval. (Enter your answer using interval notation.) (-1)" + 1(x – 2)" Σ f(x) = n = 1 (a) n2" f(x) (b) f'(x) (c) f"(x) f(x) dx| f(x) dx Assume f(x) is continuous and positive everywhere, and that Ji converges. Find the following. lim f(x) a) Assuming it exists, r+0 lim f'(x) b) Assuming it exists, 2-0 Decide on the convergence or divergence of the following ("cannot be determined" is also an option). Defend your conclusion. | f(x) dx 10 f (x + 3) dx | (f(x)+ 3x) d f(x) dx | f(x) sin r daf(x) = n2 n = 1 Find the intervals of convergence for f. (Enter your answers using interval notation.) Find the intervals of convergence for f'. Find the intervals of convergence for f".