give an example where f is not derivable at [0, ∞), but is uniformly continuous

Q: If f(x) is an integrable function defined on the closed interval [a, b] ], ƒ (x) dx represents the…

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Q: True or false ? I are Continuous on [aib] then f and g

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Q: Assume thatf is continuous everywhere. Determine whether the statement is true or false. Iff has a…

A: Assume that f is continuous everywhere  We have to determine whether the statement is true or false…

Q: 2. Recall that we proved f(r) = VI is continuous on its domain [0, 00). Explain why f(r) is…

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Q: Define f : Rm \ {0} → R by ƒ(x) = T: Prove that f is uniformly continuous on R" \ B1(0), but not…

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Q: Let BCR be the open ball of radius one centered at the origin, i.e. B = {(x₁,...,xn) € R" : x² + +…

A: As per the question we are given a n dimentional open ball B of radius 1 centered at the origin. And…

Q: Let f, (x) → f (x) on [a, b] and both Reimann Integrable. Then f, (x) uniformly converge to f(x) on…

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Q: 24. Show that f(x) = mx + b, where m,b e R, is uniformly continuous on R. %3D

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Q: Is f continuous at (0,0)? Is f continuous everywhere?

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Q: If f(x) is uniformly continuous on R, then f is bounded on R. True False

A: To say True or False

Q: For a continuous function f(x) on the set 2 [- 5,2], the net area Tx)dx must be -5 always positive…

A: The given question has been solved in detail

Q: Let ƒ(x, y) =(x2-y2)/(x2+y2)  for(x, y) ≠ (0, 0). Is it possible to define ƒ(0, 0) in a way that…

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Q: Let f(x) = x2, for all x E R. Is f € C(R)? Is f uniformly continuous on R? Justify your conclusions.…

A: Solution

Q: Suppose f : [a,∞) → R is continuous and f(x) → 0 as x →∞. Prove that f is uniformly continuous on…

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Q: (6) Are all continuous functions on ID differentiable on D?

A: we have to identify the given statement is true or not

Q: If f(x) is integrable on [a, b] , then f(x) is continuous on [a, b] . O True O False

A: Given statement:If fx is integrable on a,b, then fx is continuous on a,b.We need to check the…

Q: • Let X be a continuous r.v; with pdf f(x) : 0 < x < 1, x, f(x) = {2 – x, 1< x < 2, 0, elsewhere.…

A: We have given that, Let X be a continuous random variable with pdf: f(X) = {  x,   0&lt;x&lt;1…

Q: Let f be continuous on the interval [a,∞). Show that if the improper integral ∫∞a ∣f(x)∣ dx…

A: If f  is a continuous function on the interval [a, ∞). If  ∫a∞f(x)dx converges, Then, we use…

Q: Let f : E → R be uniformly continuous. Prove that |f| is uniformly continuous. Is the converse true,…

A: Let f: E → R be uniformly continuous.  Now, we need to show that |f| is uniformly continuous.  Now,…

Q: Let fn (x) uniformly converge to f(x) on [a, b]. If f„(x) E R[a, b], then f(x) E R[a, b] O True…

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Q: Let f:X→R be a Lipschitz continuous function defined on X. Show that f is uniformly continuous on X.

A: Given : function f:X -&gt; R is a Lipschitz continuous function defined on X . To show : f is…

Q: Every function which has counatable discontinuous points on [a,b] is Riemann integrable on [a,b]. *…

A: See the solution below.

Q: Every function which is Riemann 0/3 integrable on [a,b] is continuous. * True O False

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Q: Let f be a function defined on R satisfying If(z) - f(y)| < 히 - J), V,yER. a. Show that f is a…

A: Given that f be a function defined on ℝ satisfying…

Q: Suppose f is Reimann integrable on [a,b]. If f (x)< 0 for all x E [a, b], then S f(x)dx

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Q: Let f: X → Y and g : Y Z be uniformly continuous on X and Y, Prove go f: X→ Z is uniformly…

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Q: Suppose that ƒ is continuous and F(x) = g> 0, g'>0 and integrable on [a, b) and that lim g(x) = I-b-…

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Q: Suppose f is uniformly continuous on (a, m] and also on (m, b). Prove: f is 1 uniformly continuous…

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Q: Suppose f> 0 is integrable on [a, b], continuous at xo € [a, b], and f(xo) > 0. Prove that f>0.

A: Given f≥0 is integral on a,b continuous at x0∈a,b and fx0&gt;0

Q: Determine whether the statement is true or false. If f is continuous on [0, o0) and f(x) dx is…

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Q: If f(x) is continuous function on la,b] than is not integrable [a,b]. on Select one: O True O False

A: Topic:- function

Q: true or false If f is integrable over [a, b] and R b a f(x)dx = 0, then f(x) = 0 for all x ∈ [a, b]

A: According to the given information, it is required to tell whether the given statement is true or…

Q: 2. Using the sequential criteria for absence of uniform continuity, show that ƒ : R \ {0} → R given…

A: Letxn=1n;  yn=12n 1n-12n=2-12n=12n→0 as n→∞

Q: Show that if f is Riemann integrable on [a,b] and f(x) ≥ 0 for all x ∈ [a,b], then

A: A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is…

Q: Let f, (x) → f (x) on [a, b] and both Reimann Integrable. Then f, (x) uniformly converge to f(x) on…

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Q: Let fand g be continuous on the closed interval l0,5], suppose that / f(x) dr=A, s(x) dr=B, /…

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Q: Prove the following result: A function f that is decreasing on [a,b] is integrable on [a,b].

A: Prove the result that,        "A function f that is decreasing on[a,b] is integrable on[a,b]''

Q: true or false  If f is continuous on [a, b], then f has an antiderivative on [a, b]r

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Q: Let f, (x) uniformly converge to f(x) on [a, b). If f.(x) e R[a, b), then f(x) € R[a, b] True False

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Q: Let f: X→Y, g:Y-Z be uniformly continuous. Then gof:X-Z is uniformly continuous.

A: The main objective is to proof that the function g∘f is uniformly continuous, i.e. for every ε&gt;0,…

Q: Suppose that f is integrable on [a,b], and that g(x) is defined for all x E[a,b] and ,exists and…

A: Given: To explain the given statement as follows, To prove that    exists and equals .

Q: Let f : [0,∞) → R. Assume that f is uniformly continuous on [0, 1] and on [1,∞). Show that it is…

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Q: All continuous functions are integrable. A True B False

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Q: Let f : [a, b] → R be a differentiable function with f′ bounded. Show that f is is uniformly…

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Q: Assume that J is continuous on [0, 3 and differentiable on (0, 3) with f (z) > 6 for each z in (0,…

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Q: Assume that ƒ is continuous on [a, b] and differentiable on (a, b). Also assume that ƒ(a) and ƒ(b)…

A: According to the given information: Assume that ƒ is continuous on [a, b] and differentiable on (a,…

Q: True o false ? If fand g g are Continuous on faib I, then

A: Please see the answer below.

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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Q: f (x) = %3D x² +1

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Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.