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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) ?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→∞0Find all possible value of x for convergence and divergence both
- 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.Let (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 ?
- Show whether or not fn(X) =sin (x/n) is pointwise convergenceConsider 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. 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
- 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) dxProvide an example or explain why the request is impossible.Let’s take the domain of the functions to be all of R. (b) A sequence (fn) of differentiable functions such that (f'n) converges uniformly but the original sequence (fn) does not converge for any x ∈ R.Let f(x) = 8 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".