• Assume f : A R continuous, and A open. Is f(A) open? • Assume f: A R continuous, and A compact. Is f(A) compact? • Assume f : A → R continuous, and A closed. Is f(A) closed?

Q: If f is differentiable at a , then f is continuous at a . True O False

A:

Q: Assume that a continuous function f:R → R is X- and µ-periodic, where A and u are two nonzero…

A: We know that between two rational number lies infinite many irrational number. But function is…

Q: Prove that f(x) = |x| is continuous at all points c in ℝ.

A: Continuity of a function can be done by many methods as,

Q: Let f be a function such that f' is continuous on [a, b] Show that a

A:

Q: Prove the following statement: Assume f is a continuous function on [a, b], k is any real number |…

A:

Q: Suppose that f(x) is a continuous function on [a , b] and F(x)= | f(t) dt a Part 1 of the…

A:

Q: Let f and g be continuous functions on the interval [a,b], such that f(x)=g(x)-1 for all asx<b.…

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: Determine whether the statement is true or false. ()dr) = lim %3D If f is continuous, then O True O…

A: We have given that f is continuous.

Q: Determine whether the statement " If f is continuous for all nonzero x and y, and f(0, 0) = 0, then…

A:

Q: 6. Provide a counter example and explain that the following statement is false. "If f is continuous…

A: We want to find a counter example of given statement.

Q: D. Prove that f(x) = |x - 6| is continuous at all points c in R.

A: Given function is fx=x−6. We have to prove that the function fx=x−6 is continuous at all point c in…

Q: a continuous function f such that f(4) = 4, f'(4) = 0, ƒ'"(x) 4 is:

A:

Q: Assume f and g are both continuous on the closed interval [a, b), f(a) g(b). Prove f (x) = g(x) for…

A:

Q: 6. Let f: R? → R? be a continuous function. Prove that f ([0,1] × [0,1]) is compact.

A:

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: f is continuous at a if and only if |f| is continuous at a.

A: False

Q: 1 If the function f is continuous on [a,b], then there exists at least one number C in [a,b] such…

A: If the function f is continuous on [a, b], then there exists at least one number C in [a, b] such…

Q: (a) Use the definition to prove that f(x) = is uniformly continuous on the x +1 domain [0, 00). x- 2…

A:

Q: Prove: If f is continuous on [a,b] and differentiable on nd f'(x) < 0, VE(a, b) then is ƒ is…

A:

Q: A function f: Z x Z is defined as f(m, n) = 3n - 4m.  Verify whether this function is injective and…

A: f: Z x Z is defined as f(m, n) = 3n - 4m So, f(m, n) = f(a, b) implies, 3n - 4m = 3b - 4a i.e. 3(n…

Q: 2: If f(x) and g(x) are continuous on (a,b]and if f =[s . Prove that 3cela,b] such that f(c)=g(c).…

A:

Q: If a function f(x) is not defined at x-a, then it is not continuous at an open interval (a, b) * O…

A: Given Statement is "False"

Q: (b) Let X = (a, b]. Construct a continuous function f : X → R (set of real numbers) which is…

A:

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: Determine if g (r)= is continuous at a = 8.

A:

Q: Let f and g be functions that are continuous on [a, b] and differentiable on (a, b). Prove that if…

A:

Q: Suppose that the function f(x) = {x“ + 1, x>-1 x <-1 is continuous on R, then a = lax, %3D 1 2 No…

A: The function is given by fx = x2+1, x≥-1ax, x&lt;-1 To evaluate: The value of a if the function is…

Q: ) Let f be a continuous function that maps a finite closed interval [a, b] into itself. Prove that f…

A:

Q: Suppose that f and g are both continuous functions. Also f, g € [a, b] and that f= fg. Prove that 3c…

A:

Q: Suppese that X, X and that f is continuous. Show that f(X,) 1(X). Show w that ETLJY

A: Since you have asked multiple questions, we will solve the first Question for you. If you want any…

Q: Determine whether the statement is true or false. If it is false, explain why or give an example…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Let f: X→Y be a function. Please prove that if f(X-A)=Y-f(A) for any A⊆X, then f is onto.

A:

Q: et f: (a, b) → R. If f'(x) exists for each x, a < x <b, prove that f is continuous there.

A:

Q: 1 - is continuous at a. f If f(x) 0 for all x in the interval I, and f is continuous at a in I then…

A: 1/f is rational function this is discontinous where either f is discontinous or f is zero  But f is…

Q: If f is a continuous function on [a, b], and if f (a) = -2 and ƒ (b) = 2, then there exists a real…

A: Hi, you can find the answer in the handwritten form.  Given f(a)=-2  f(b)= 2

Q: Let f be a continuous function defined on E. Is it true that f^-1(A) is always measurable if A is…

A: Prove the given statement

Q: x + 6|, if x - 1 x² – 3x

A:

Q: 4. Let A := (0, 1] and let f : A → R be defined by f(x) = !. Prove that f is continuous on A.

A:

Q: Give a 8, e proof that f(x) = x² + 3x is continuous at x = a, where a is any real %3D number.

A:

Q: f, g: A → R be continuous on A, and let h(x) Let = max{ f(x), g(x) } for x belong to A. (a) Show…

A: Given, f, g: A→R be continuous on AAndhx=maxfx, gx for x belong to A.

Q: Let A := (0,1] and let f : A → R be defined by f(x) = . Prove that f is continuous on A.

A: Consider the provided question,

Q: Suppose f : [a,b] → R is a continuous function. Prove that the direct image f([a,b]) is a closed and…

A:

Q: Consider the function f : [a, b] → (a, b] with ƒ being continuous. Show that there is a number c in…

A:

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: Prove that f(x) = x 3 - l2x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Given information: The function is f(x)=x3-x. Calculation: We need to find the value of f'(x) and…

Q: Suppose that a function y = f(x) is continuous on I = [a,b] and if the set f(I) = [f(a),f(b)], then…

A:

Q: Let f,g: D -> R be conitnuous at c ∈ D. Prove that fg is continuous at c.

A:

Q: 3. Let f(r) = 8 – 4x4/5. Carefully explain why f is continuous on [-1, 1), f(-1) = f(1), but there…

A:

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.