Question 2. Let f: (a,b)R be a continuous function and let cE (a, 6). Suppose that f is uniformly continnous on both (a,c) and (c, b). Proe that f is uniformly continuons on (a, b).

Q: Question 4 Rolle's theorem states that if a function fis continuous on the closed interval [a, b]…

A: Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and…

Q: If f is continuous function on la, bl, then zf(z)dz = = (z)dz. Select one: a. True b. False

A:

Q: Evaluate the function shown, where ƒ is continuous on [3, 8], ƒ(3) = 4,and ƒ(8) = 20.

A:

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.

Q: If a continuous RV X follows N(0, 2), find P{1 1}.

A:

Q: If f(1) = 2, f(4) = 7, f'(1) = 5, f'(4) 3, and f"(x) is continuous, what is x f"(x) dx? %3D

A: Given: f(1)=2f(4)=7f'(1)=5f'(4)=3I=∫14xf''(x)dx=?

Q: (c) If f(x) is continuous on a closed interval [a, b] and f(x)dx > 0, then f(x) > 0 for some x E [a,…

A: Since you have asked multiple questions in single request.so we will answering only first question.…

Q: Question.1 Let f: [a, b] → R be a continuous function. Prove that fE R. Is the converse true? Why?

A:

Q: Question 3. that is differentiable on (1, 2). If f(1) = 0 and f'(x) < 10, show that f (2) is bounded…

A:

Q: If f is continuous on [a,b], th then (A) True B False [Ttf(x) dx = π * f(x) dx

A: The solution is given in the next step.

Q: f and g are continuous on la,b], then F(x)+ g(x)]dx= | f(x)dx+ g(x)dx A) True B) False

A: the integral of sum of two or more functions is equal to the sum of their integrals.

Q: Theorem 10 Let (a, b] c R and f : [a, b] → R be a function of bounded variation on [a, b]. Then f is…

A:

Q: and g are continuous on [a,b] and f(x) > g(x) for all x €[a,b] then f(x)dx> ] g(x)dx a

A:

Q: Show that a function f : R? → R defined by S (7, y) = ! if (x, y) + (0,0) 0, if x = y = 0 is…

A:

Q: =f and g are continuous on [a,b], then [*[f(x) + g(x)]dx = f(x) [ √° g(x) f(x)dx+ Sº g(x)dx (A) True…

A:

Q: 1. a. Suppose F is of bounded variation on [a,b], then |F'| = VF' a.e b. F is absolute continuous…

A: It is given that Suppose F is of bounded variation on [a,b]

Q: 1. Show that if f [a, b] → R is increasing, then f is of bounded : variation.

A:

Q: If f is continuous function on a, b|, then Sæf(x)dæ = x J° f(x)da S. f(x)dx :Select one a. False b.…

A:

Q: Let f(z, y) be a continuous function such that f2, Su Jay and fuz are all continuous everywhere. If…

A:

Q: Theorem 10 Let (a, b] c R and f : [a, b] → R be a function of bounded variation on fa, b). Then f is…

A:

Q: Question 6. For what value of b is x, x -2 continuous at every x?

A:

Q: If a function f: D R is differentiable at a point (a, b)ED, then f is continuous at (a, b);

A:

Q: 1. Show that if f [a,b] → R is increasing, then f is of bounded variation.

A: Given f: a, b → ℝ is increasing function.

Q: 6. Let f: A R be a continuous function. Show that if A is compact then sup f(A)e f(A).

A:

Q: Theorem 12. Let f be a non-zero continuous function on R such that f(x+y)=f(x) f(y) ¥x, yER. Show…

A:

Q: Suppose f and g are two continuous functions on [a,b] and that f(a) g(b). Show that f(x) = g(x) for…

A: Suppose fand gare two continuous functions on [a, b] and f(a) &lt;g(a) but  f(b) &gt;g(b).  Show…

Q: اعه W = {4; [0₁+0) IR is absolutely continuous what is absolutely continuous ?

A:

Q: It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function…

A: To solve an equation fx=0, if f is continuous on an interval a,b such that fa and fb are of opposite…

Q: Theorem 10 Let (a, b] c R and f: [a, b] → R be a function of bounded variation on [a, b]. Then f is…

A:

Q: Let f(x) = 1, for all x E (0,1). Is f continuous on (0, 1)? Is f uniformly continuous on (0, 1)?…

A:

Q: Suppose f and g are continuous functions on the interval [a, b], then SEAFX) – g(x)) dx = [." f(x)dx…

A:

Q: 1. Suppose a increases on [a, b], a<xo Sb, a is continuous at xo, f(xo) 1, and S(x) 0 if x xo. Prove…

A:

Q: 10. If f(x) and F(x) then f(2x)dx = are continuous functions such that F'(x) = f(x) on R, %3D %3D

A: We have to solve given problem:

Q: 4. Let f:A→R and g:B→R where f(A)cB. Explain to prove that if f is continuous at CE A and g is…

A:

Q: For what value of the constant c is the function f continuous on (- o0, 0) where S (1)² – c if y 2

A:

Q: Isg continuous on (0,1]? * if x 4.

A: Solution:Given  g(x)=xif x≤03if 0&lt;x≤13-x2if 1&lt;x≤4x-3if x&gt;4To find:g is continuous at (0,1]

Q: 1. "Let fE a[a, b]. For x E (a. b), set F(x)=ff. Prove that F is continuous on [a, b].

A:

Q: If f(1) = 2, f(4) = 7, f'(1) = 5, f'(4) = 3, and f"(x) is continuous, what is xf"(x) dx? %3D %3D

A: In the question it is asked to solve for the initial value integration given in question.

Q: If f is continuous and even on [-a, a], then f(x S'f)dx = [ °ft-x)dx Select one: O True False

A: f is continuous and even on [-a,a] , then ∫-aaf(x)dx=∫-aaf(-x)dx

Q: If f is continuous on [a,b], then [a,b], then A) True B) False [ f(x)dx = πf f(x) dx

A:

Q: If f is continuous on [a,b], then f(x)dx= π + A) True B) False f(x) dx

A: Given that f is continuous on a,b. ∫abπ fx dx = π ∫abfx dx Say that the statement is true or false.…

Q: Let D := [0,1] and let ƒ : D → R be the function defined by f(x) = Vr. Show that f is uniformly…

A: Consider the provided question,

Q: 1. Show that if u = x – y, v = y, then for any continuous function f, -" f(r – y, y)dydæ e=s(u+v)…

A:

Q: If ƒ : (0,1] → R and g : [1,2) → R are uniformly continuous on their domains, and S (x) for æ € (0,…

A: The given function is uniformly continuous in the intervals 0,1 and 1,0. The interval 0,2 is just a…

Q: 1. Suppose f(r) is a continuous function on r € [a, b]. Prove f(x)dr < S°S(x)| dx.

A:

Q: Question 9. (a) Given a continuous function f: RR and a connected subset S € R, is f¹(S) connected?…

A: Thank you for the question. As per guidelines, we will solve the first question. Kindly repost the…

Q: 1. Given that f is continuous on a, b|, compare f(b)(b – a) | f(x) dx, and if f is constant on a,…

A:

Q: Suppose that f is a continuous, strictly increasing function on [0, 1] and that the •1 range of f is…

A: Here given function is continuous and strictly increasing function on [0,1]

Q: Find the value k that makes the function continuous. S ka?, f (æ) = x > 2 ka + 4, г 2

A: Given thatfx=kx2         x&gt;2kx+4    x≤2We have to find value of K for which function is…

Q: If f is continuous function on la, b, then f(z)dr = z f(x)dz. Select one: a. True b. False

A: we have been given that the function f is continuous on [a, b].

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.