Let f(x) be a given continuous function on a closed interval [a, b]. Use the Extreme Value Theorem to prove that for a givenr > 0 there exists 0 > 0 so that for all a

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

A: If a function fx is differentiable at x=a then function must be continuous at x=a. But a function…

Q: If f is continuous at a number, x, then f is differentiable at x. True

A:

Q: f(x+y)=f(x)+f(y) and f is continuous at 0. Prove that f is continuous at any a.

A: To show that the continuity at x=0 implies continuity at all x (under the given condition on the…

Q: Sketch the graph of a continuous function y = h(x) such that a. h(0) = 0, -2 < h(x) < 2 for all x,…

A:

Q: Which of the following statements are true for f(x)? f(x) always has a critical point on [a, b]…

A:

Q: Consider f(x) = 3|x|, which is in R[-2, 2]. a. Find a continuous function F(x) such that F'(x) =…

A:

Q: Let f(x) = { *, if x= 2 ; for the eax if x + 2 function to be continuous at x = 2 the value of a is

A:

Q: Prove that if f is a continuous function on a closed interval [a, b], then

A: We have to prove the given inequality for a continuous function on a closed interval [a,b].

Q: If f is differentiable at a number, x, then f is continuous at x.

A:

Q: c =

A:

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: Suppose that f is a function on the interval [1, 3] such that –1 < f(x) < 1 for all x and f(x) dx…

A: It is given that ∫13fxdx=0 and -1≤fx≤1.

Q: c. Show that f(x) is NOT uniformly continuous on [0, +∞).

A:

Q: Let g : R → R be a bounded function and define f : R\{0} →R by 1 f(x) = g(x)+ Using only the…

A: It is given that ;                f : R \ 0 →R  defined by     fx = gx + 1x2 Where,   gx  is a…

Q: Prove that f(x) =√x is uniformly continuous on [0,∞).

A: According to the question, we have to prove that the function fx=x is uniformly continuous on [0,∞).…

Q: If f : R+ → R is a function such that f(x + y) = f(x)f(y) for all x,y ∈ R+. Prove that f is…

A:

Q: If a function f(x) is not defined at x-c, then it is not continuous at a close interval [a, c). * O…

A: We know that function is continuous at c if the 1) f(c) is defined 2)limit of f(x) exist, Here…

Q: Determine the domain of the function and prove that it is continuous on its domain f(x) = e - x 2

A: Consider the given function. fx=e-x2 The function is define for every real value of x. Hence,…

Q: Find all values of "c" such that the function f(x) is continuous over  ( − ∞ , ∞ ) f(x) = 1 - x^2…

A:

Q: Suppose f : R → R is a continuous function with the property that f(x) = 0 for every x E Q. Prove…

A:

Q: Sketch the graph of a continuous function y = h(x) such that a. h(0) = 0,–2 < h(x) < 2 for all x,…

A:

Q: Given a continuous function of which is twice differentiable, if f" (c) > 0, which of the following…

A:

Q: Use limits to determine if f(x) = 1 is continuous at x = 0. (z+2)?

A:

Q: Set f(x) = xª for x ≥ 0 and f(x) = 0 for x < 0. For which values of a is f continuous at 0, and for…

A:

Q: Assume that ƒ is continuous on [0,1] and that f(0) = 2 and f(1) = 1. Show that the graph of y = f(x)…

A: given  f is continuous on [0,1] f0=2,f1=1y=x2+1

Q: Show that the limit of the function f(x,y) = at (0,0) does not exist. (x+y)² x² + y²

A: Limit of a Function: Like a single-variable function, a multi-variable function can also have limit.…

Q: Prove that the function f(r) = is uniformly continuous on (0, 00) エ十2

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: 4. For a function f:D → R and a point xo in D, define A = {x in D x ≥ xo} and B: {x in D x ≤ xo).…

A: Recall definition of conituity as, Right hand limit = left hand limit = value of function at the…

Q: show that f,given by f(x) x +0 )is continuous on R-{0}

A:

Q: Let f(x) = 0 if x∈Q if x and f(x) = 1 if x ∈R\Q. Prove that y=f(x) is not continuous at any point…

A:

Q: Use limits to determine if f(x) = is continuous at x = 1. z-2

A:

Q: Use the definition to prove that the function in f(x) = 2x - 3 is uniformly continuous.

A:

Q: Find the value/s of a such that the function f is continuous everywhere.

A:

Q: Find an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) → 0…

A: To give an example of a function f : [−1,1] → R such that for A := [0,1], the restriction f |A(x) →…

Q: Let g(x) be a continuous function for all æ E R. If f'(x) = g(x)f(x) for all a E R, find f(x). Note:…

A:

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: If f is a continuous function and F'(x)=f(x) for all x € R, then - x)dx=?

A:

Q: If f is differentiable at a ,then f is continuous at a

A: Every differentiable function is continuous.

Q: For the function f(x, y) = xy((x2 − y2) /( x2 + y2) ) define f(0, 0) such that f is continuous at…

A: A function f is continuous at the origin, the function value at the origin must be equal to the…

Q: For every continuous f:[0,+0) →R _function , if a>0 2fx'f(x*)dx =[ xf(x)dx Demonstrate that this…

A:

Q: If a function f is continuous at 0, show that g(x) = xf(x) is differentiable at 0 and find the value…

A:

Q: (2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies…

A:

Q: Find a constant b such that h is continuous at x = 2, where x +1 for |x| 2 With this choice of b,…

A: It is given that the continuous function  hx=x+1for x&lt;2b-x2for x≥2 It is also given that function…

Q: Every continuous function on R satisfies the conditions of MVT. * False True

A: We have to evaluate that the statement, ''Every continuous function on R satisfy the conditions of…

Q: Show that f : (0,∞) → R defined by f(x) := 1/x is not Lipschitz continuous.

A: Definition:  Let S⊂ℝ and a  function f:S→ℝ is Lipschitz Continuous at x∈S if there is a constant C…

Q: Consider the piecewise-defined function Ax if x 0 h(x) е (a) Give values for A and B so that h(x)…

A: Given Data: Function: hx=Axif x≤0e-1/x+Bif x&gt;0 (a) The condition for the continuous function is,…

Q: Let f : R → R be a continuous function satisfying f(x + y) = f (x) · fG) for all x, y E R 1. Show…

A: solution of above question is given below

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.