Define a function f(x) such that all of the following conditions are satisfied: • The domain of f is R, which means that f(x) is defined for all x in R,

Q: Prove that f(x) = x 3 - 12x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Since, we know  that a polynomial function is always differentiable.⇒f(x)=x3-12x-(i) is always…

Q: Prove that the function given by f(x) =x3-3x2+3x-100 is increasing in R

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Q: Define functions H and K from R to R by the fol- lowing formulas: For every x ER, H(x) = (x– 2)² and…

A: Observe the given functions as follows.

Q: Verify that the three conditions of the Rolle Theorem hypothesis hold for the function f (x) = 1-2x…

A: Three conditions of the Rolle's theorem hypothesis. Consider function f(x), it should be continuous…

Q: 2. f(x) = [-3,1], k 2 x+2

A: Intermediate value theorem  A function f(x) is said to satisfy intermediate value property if in…

Q: Determine the domain of the function and prove that it is continuous on its domain using Theorems…

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Q: Prove that f(x) = (x – 1)2 + (x + 1)? is an even function. A function k(x) is an even function if…

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Q: Suppose f is a function which satisfies the properties: f(x + y) = f(x) + f(y) + x²y + xy? for all…

A: Given that, f(x+y)=f(x)+f(y)+x2y+xy2 for all real numbers 'x' and 'y'  and, limx→0f(x)x=1   As we…

Q: Prove that the function f(x) = x3-3x2+3x-100 is increasing on R

A: Given function is : f(x) = x3-3x2+3x-100 We have to prove that f(x) is increasing on R

Q: Define functions F and G from R to R by the following formulas: F(x) = (x + 1)(x – 3) and G(x) = (x…

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Q: There exists a function f such that f(x) > 0, f(x) 0 for all x. True False

A: We are entitled to solve only one question at a time so, I am providing you an answer of the same.…

Q: For any k > 0 E Z, show that there is a function f : Z* → Z which is both one-to-one and onto. (Here…

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Q: Consider the function x - 3 for x4 By using the characterization of continuous function in terms of…

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Q: A function f is ___________ on an interval I if, for any choice of x1 and x2 in I, with x1 < x2,…

A: To fill: The blank of the statement. Given: The statement is "A function f is _______ on an open…

Q: Suppose that ƒ : [a, b] → R is a continuous and one-to-one function. Prove that ƒ is strictly…

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Q: Suppose that f: R (0, ∞) and that f'(x) = f(x) ‡ 0. Prove that (f-¹)(x) = 1/x for x > 0.

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Q: Suppose f is a function which satisfies the properties: f(x + y) = f(x) + f(y) + x²y+ xy? %3D for…

A: Given that: fx+y=fx+fy+x2y+xy2limx→0fxx=1   Concept used: Derivative of fx is given by…

Q: Define functions f and g from R to R by the fol- lowing formulas: For every x E R, 2x° + 2x = 2x and…

A: It is given that, f(x) and g(x) are defined from R to R by the following formulas:

Q: Prove that if (c, f(c)) is a point of inflection of the graph of f and f" exists in an open interval…

A: It is given that (c,f(c)) is a point of inflection of the function f.

Q: Suppose that f : [a,b] → R is a continuous and one-to-one function. Prove that f is strictly…

A: We will prove this by contradiction.

Q: Consider the following statements 1- The function f defined by f(x) = x³ is decreasing in R 2- If a…

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Q: Suppose f and g are contimuously differentiable functions such that f(x) = d(x) and g(x) f(x) and…

A: Define a new function                                                   r(x) = f2(x) -…

Q: Prove that if f (0) = g(0) and f' (x) ≤ g' (x) for x ≥ 0, then f (x) ≤ g(x) for all x ≥ 0. Hint:…

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Q: Let f(x) = { 3, 1< x < 2 0, 2 < a <5 and Let F(x) = { 3x, 1<x < 2 6, 2 <x < 5 Show that wether the…

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Q: Find the value of that will make the function f(x) continuous everywhere. f(x) = {ka² -2, (4x+k,…

A: Given function is- f(x)=4x+kx≤-5kx2-2x&gt;-5 To find- To find the value of k that will make the…

Q: Prove that if f(x) is a differentiable function everywhere and f'(x) # 1 for any value of x, then…

A: The objective here is to prove that the mentioned function f(x) has maximum one fixed point. Given:…

Q: Let g: Z x Z Z x Z be the function given by g(x, y) = (3x +y, x- y). Find, with proof, an element of…

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Q: Suppose that f: R→R is everywhere differentiable and that f'(x) 1 for all real numbers x. Prove that…

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Q: Assume that f(x) is a function defined by x²-3x+1 f(x) = 2x - 1 for 2 ≤ x ≤ 3. Prove that f(x) is…

A: note : As per our company guidelines we are supposed to answer ?️only the first question. Kindly…

Q: Let f be differentiable on an interval A. If f'(x) not equal to  0 on A,show that f is one-to-one on…

A: Given that, Let f be differentiable on an interval A. If f'(x) not equal to  0 on A. We have to show…

Q: Define g(-1) in a way that extends g(x) = 1 to be continuous at x= - 1. x² -x-2 2 9(-1)-D

A: Given that: g(x)=x2-1x2-x-2

Q: Assume that f and g are differentiable functions such that f (0) > g(0) and that for all x E R,…

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Q: Let f:Z→Z be defined by f(n) = 2n+ 1 for each n∈Z and g:R→R defined by g(x) = 2x+ 1 for each x∈R.…

A: Given function is Let f:Z→Z be defined by f(n) = 2n+ 1 for each n∈Z and g:R→R defined by g(x) = 2x+…

Q: Find x for which f(x) is not continuous: 3 x-1 f(x) =-+ x'x2 -1

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Q: Suppose that f: R R is everywhere differentiable and that f'(x) 1 for all real numbers x. Prove that…

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Q: assume f(g(x))=x and f'(x)=1+f2(x) for all x. show that g'(x)=1/(1+x2)

A: Differentiate both sides of f(g(x))=x with respect to x. Use chain rule on the left side.

Q: 4. Let f:R→ R be a function defined by f(x)=x for every xeX . Show that f is a continuous.

A: Let A ⊆ R , let f:A → R and let c ∈ A. We say that f is continuous at c if, given any number ε&gt;0,…

Q: If f(x) and g(x) are defined on domains A and B respectively, then the domain of f(x) + g(x) is? *

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Q: Any function f defined on R should have a relative extremum.   True or False

A: In the question we have to find whether the statement is true or not.

Q: a. f(x) x-2 x-7 b.f(x) = r2+1

A: (a) the given function is f(x)=1x-2 the function is continuous in interval [a,b] if it is continuous…

Q: If we apply Rolle's Theorem to the function f(x) = 2x² - 8x - 1 on the interval [0, 4], how many…

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Q: Let f(z) be a real valued function defined for satisfying f(1) = 1 and f'(2) = then lim f(z) =)…

A: Given function is, f'x=1x2+fx2 The above equation f'x is always positive which means fx  is always…

Q: Determine if the function f(x) = 6/1 – x² satisfies the Mean Value Theorem on [0, 1]. If so, find…

A: givenf(x)=61-x2interval is [0,1]according to mean value theorem f'(c)=f(1)-f(0)1-0

Q: (i) Let ƒ be the function defined on (-o, 0) by f(x) = r$(3 – x)}. (2 – x) -2 Given that f'(x) = and…

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Q: If f is C2 on an interval prove that f(x + h) – 2f(x) + f(x – h) lim h-0 = f" (x). h2

A: We can prove this with the help of Lagrange Mean value Theorem as follows.

Q: Suppose f and g are functions with domain R. If both f and g are even but ƒ +g is odd, then prove…

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

Q: Let f(x) be the function defined by f(x) = x3 -1 for x < −1, f(x) = ax + b for −1 ≤ x ≤ 4, and f(x)…

A: Given f(x) x3-1   for x&lt;-1        ax +b  for -1≤x &lt;4        x2+2 for x≥4

Q: 1. f(x) = x² + 5x – 6; [-1,2], k = 4

A: Intermediate value theorem

Q: Define g(3) in a way that extends g(x) = (x2 - 9)/(x - 3) to be continuous at x = 3.

A: Here, the given function The function gx is continuous at  x=a, if the following holds: (i) ga is…

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.