7. Prove that if f and g are continuous then so are max(f, g) and min(ƒ, g).

Q: if and only if F-1: R(F) →X is bounded.

A:

Q: Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). Which of the…

A: Given: Suppose that Q(x, y) is a function such that 1/Q(x, y) is continuous for all (x, y). To Find:…

Q: 3. If f is continuous and | f(x) dx = 10, find af(x*) dr. Give proper reasoning.

A:

Q: 4. Use the definition of uniformly continuity to prove that f (x) 1- x is uniformly continuous on…

A:

Q: If f(x) is continuous on the interval [a, b] and a <c < b, | f(x) dæ is equal to then

A:

Q: if f:[0,2] 3 R such that if x>0 f(x) = { ²2 if x<0 show that whether f is continuous in [0,2]

A:

Q: Suppose that ƒ is continuous in [0,1] and that f(x) E [0, 1] for all x. Show that f(x) = x for some…

A: Given that f:0,1→0,1 is continuous on 0,1. To show fx=x for some x∈0,1

Q: If f(x) is continuous everywhere, then If(x)| is continuous everywhere. TRUE O FALSE

A: If f(x) is continuous everywhere, then |f(x)| is continuous everywhere.True of false.

Q: Which of the following is at least piecewise continuous in the interval [0, 10]. 2t, 0 st<1 2t - 3 ,…

A:

Q: If  and f is continuous, then  f must have at least one zero between  a and b. (Assume a ≠ b.)

A: Case1: Whenfx=0 , then ∫abfxdx=0 is obvious.

Q: [0,1] 6. If f is continuous on • prove that

A: We can solve an integral of some function f(x) w.r.t. x by using substituting, say, t=f(x) which…

Q: 2. Show that if f : R → R is continuous on R and f(r) = 0 for all r E Q then f(r) = 0 for all x E R.

A:

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

A:

Q: If f is continuous on [a, b], then the horizontal line y = k for every k between f (a) and f (b)…

A: If function 'f'  is continuous on closed interval [a, b], and if f(a)≠ f(b), then for any number k…

Q: Prove that the function ƒ : (0, ∞) → R defined by ƒ(x) = x³/² = (√x)³ is not uniformly continuous.…

A: To Prove: The function f:0, ∞→ℝ defined by fx=x3/2=x3 is not uniformly continuous. To Use the…

Q: Prove directly from the definition that if X and Y are independent and f and g are measurable…

A: Solution: Given X and Y are independent and f and g are measurable function then we have to prove…

Q: Suppose that a function ƒ is continuous on the closed interval [0, 1] and that 0&lt;=ƒ(x)&lt;= 1 for…

A: Let us assume a function g(x)=f(x)-x Since both f and x are continuous on the closed interval [0, 1]…

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: Let f : X 'n Y be a continuous function. If X is disconnected then so is f(X). True False

A: As image of a connected set is connected in of every continuous function.

Q: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

A:

Q: If f:X - Y is a continuous function then f- (V) is open in X for every open set V in Y. True False

A:

Q: 8. Prove that a function which is uniformly continuous on X is also continuous on X

A:

Q: Suppose that the function f: [0, 1]  in R is continuous and thatf (x)&gt;= 2     if 0 &lt;= x &lt;…

A:

Q: Let f(æ) = In(z') f'(2) =

A:

Q: Suppose h is a function such that h(2) = -2, h'(2) = 2, h"(2) = 4, h(5) = 5, h'(5) = 7, h"(5) = 12,…

A: We have to solve

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

A:

Q: 7. Show that if f is differentiable on (a, b), and f'(c) > 0, where ce (a, b) and if f' is…

A:

Q: Suppose h is a function such that h(2) = -2, h'(2) = 2, h"(2) = 4, h(5) = 5, h'(5) = 7, h"(5) = 12,…

A:

Q: (2х + 6, хS \x² + 3, x > Finition of continuity. f(x) =

A: Given,                  f(x) = 2x+6x≤0x2+3x&gt;0Now,                LHL = 6 And        RHL = 3Here,…

Q: 2: A necessary condition for a function to be unimodal on the interval [a, b] is that strictly…

A: Statement: A necessary condition for a function to be a unimodal on the interval [a,b] is that…

Q: 4. Suppose that f is continuous on a closed interval [a, b] and differentiable on the open interval…

A:

Q: Suppose that f(1) = 4,| f(4) = –6, f' (1) = 6, f'(4) = 3, and f" is continuous. Find the value of 1…

A: Given: I=∫14xf"xdx, f1=4, f4=-6, f'1=6, f'4=3 For evaluating given integral, we will use integral by…

Q: 8. PROVE: Suppose that SCR is connected, and that f : R → R is continuous. Then f(S) is connected.…

A:

Q: If functions f(x) and g(x) are continuous everywhere, then (f/g)(x) is also continuous everywhere.…

A: If f(x) and g(x) is continuous everywhere  Then    f(x)/g(x) is  also continuous every where  only…

Q: Suppose that ci < c2 and that f takes on local maxima at ci and c2. Prove that if f is continuous on…

A:

Q: 4. Suppose that f and g are continuous on the closed interval [a,b] and dif- ferentiable on the open…

A: That's easy, you just need to use Lagranges mean value theorem and given conditions and then the…

Q: If f is continuous on [a, b] and cE[a, b], then f(x)dx does not exist. Select one: True False

A: Topic:- definite integration of continuous function

Q: Suppose that f(1) = 5, ƒ(4) = -8, f'(1) = 4, f'(4) = -2, and f" is continuous. Find the value of |…

A:

Q: 9. (True/False) If f(x,y) is a continuous function on a closed, but unbounded set D, then f(x,y)…

A: False     because there are some functions which are continuous on a closed, but unbounded set,…

Q: 2. Show by definition that the function f : R R, f(x) = ||, is continuous on R. Hint: you may need…

A:

Q: 3. If f is continuous and | f(2) dr = 10, find | af(x*) dr. %3D Give proper reasoning.

A: It is given that if f is continuous and ∫09f(x)dx=10 We have to find ∫03xf(x2)dx=?

Q: 5. Suppose f : [a, b] → [a, b] is continuous. Prove there exists c E [a, b] such that f(c) = c.…

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Q: 2.23 Let 1 - x2 f(x): if x € Q, if r ¢ Q. Find all points p at which f is continuous. Prove your…

A:

Q: 6. Show that if f(2) is continuous on a domain D, and if f(z)$ is analytic on D, then f(z) is…

A: Given :- If fz is continuous on a domain D. To Proof :- If fz8 is analytic on D , then fz is…

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

A:

Q: If f : R + Ris continuous and periodic, prove that f is uniformly continuous.

A: Given f:R→R is a periodic function with period k that is f(x+k)=f(x) To prove: if f:R→R is…

Q: Let S(x²+2x)², x> 1, f(x) = Tax+ bx², ах x< 1. Compute for all a and b such that the function f(x)…

A: Proceed as shown

Q: Suppose f(x) is a bounded real - valued function on [a.b) such that fela,b] . Does it follows that…

A: A function is bounded in [a,b] if it can't achieve value beyond a and b. i.e.maximum acheivable…

Q: 3. (a) Prove that iff is continuous everywhere, then |f| is continuous everywhere.

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.