Show that f is strictly monotonic on the given interval and therefore has an inverse function on that interval. f(x) = ∣x + 2∣, [−2, ∞)

Q: find F as a function of x and evaluate it at x = 0, x = π/4, and x = π/2.

A: The given function is Fx=∫0xcosθ dθ . This impulse that, Fx=∫0xcosθ dθ=sinθ0x=sinx-sin0=sinx-0=sinx…

Q: Find the the domain of the function f(x) x2 + 4x 45 - O {x | x -9 and x # 5} O(XI x # -45} OXI x -7}…

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Q: Find f(x) if f°(x)=cos x with f(0) =1, f'(0) = 2 , and f'(-)=3.

A: Keep on integrating we are end up with an answer. For those constants we need these initial…

Q: Show that the function f (x) = 1- x is bijective and also find (i) its inverse function.

A: Bijective Function :  A function is said to be bijective or bijection, if a function f: A → B…

Q: Let f(x) = x2 / (x2-1) . (a) From a graph of f, explain why f is not invertible. Furthermore,…

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Q: let a function f(x ) be given by f(x) = log(n+x) log(e+x) prove that the f(x) is decreasing on…

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Q: Let [x – 1 when x 1 (a) Is f increasing? (b) Does f have an inverse? If so, what is it?

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Q: 3. If the function f : R R is defined by the rule f (x) = 4x + 1 for all real numbers x is…

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Q: iff.g,h becontinuous functions on [0,a]such that and sh(x)-4h(a - x)=5, then provw that…

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Q: Find f g and g of, where f (x) = x^2 + 1 and g(x) =3x + 2, are functions from R to R.

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Q: Let f(x) = log, (x). %3D Select all the correct g(x) functions such that f (x) is O(g(x)). g(x) = x

A: These problems following by carefully applying the definition of the Big O notation. The typeset…

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: If f and g are both odd functions, is their product necessarily odd, even or neither? Prove it

A: Given data: *f and g are both odd function. By the definition of the odd function if a function f(x)…

Q: Let f(x) = (x – 1)-2 Show that f(0) = f(2) but there is no number c in (0,2) such that f' (c) = 0.…

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Q: Let f and g be the functions defined by f : R → R+ ∪ {0} with f (x) = x2 and g : R+ ∪ {0} → R with…

A: (fog)(x) is the value of the function f(x) at g(x) (fog)(x)= f(g(x))

Q: Let f : X → Y be an invertible function. Show that f has unique inverse.

A: Given

Q: snip

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

Q: Show that f is strictly monotonic on the given interval and therefore has an inverse function on…

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Q: Determine the status of the function f(x) x² + 1'* € (-0, co0). strictly increasing O

A: NOTE: Refresh your page if you can't see any equations. . take derivative

Q: For any function f(r), if g(x) = f(x²) then g(z) is always increasing.

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Q: For a nonnegative interger n, let the function fn be given (se picture) n! = 1 * 2 * * * n, with…

A: We will first prove the equation by induction. Then, we show how to determine whether inverse exists…

Q: Let f: X → Y be an invertible function. Show that the inverse of f-1 is f, i.e.,(f-1)-1= f.

A: To show the inverse f-1is f.

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: Let h(x) = In[(f(x))² +1. Suppose that f(1) = -1 and f' (1) = 1. Find h'(1).

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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: Let f (x) = x and g (x)=|x| be two real-valued functions, O (x) be a function satisfying the…

A: Introduction :

Q: Let h(x)=f(x)/g(x). If f(x)=−cos(x) and g(x)=4x2+2x−3, what is h′(x)?

A: If f(x)=−cos(x) and g(x)=4x2+2x−3, what is h′(x)?

Q: Show that f is strictly monotonic on the given interval and therefore has an inverse function on…

A: Show that f is strictly monotonic on the given interval and therefore has an inverse function on…

Q: Determine the domain of the function f(x) = In(ln r) + x

A: f(x) = ln(ln x) + x f(x) is not defined if ln x <= 0 [as then ln(ln x) is not defined] ln x <=…

Q: Show that the function f(x) =x^-4 + cos (-2x) is even, odd, or neither.

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Q: Suppose thatf and g are functions that are differentiable at x = 1 and that f(1) = 2, f '(1) = -1,…

A: Given :hx = x2+1gx

Q: Given a surjective function f: [-1, 1] → [-1, 1], prove or disprove: f-(f{}) = {0}. Find f(E) and…

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Q: Let f(x)={tan(2x) ax+b if x>0 if x<0 For which values of a and b is f(x) differentiable for every…

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Q: Show that if f(x) is a non-negative, increasing concavefunction, then ln [ f(x)] is also a concave…

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Q: 10. Consider the function if a E Q -a? if x ¢ Q. f(x) = Prove that f'(0) = 0. %3D

A: Solution:Given  f(x)=x2if x∈Q-x2if x∉QTo prove f'(0)=0Formuladdx(xn)=nxn-1

Q: Prove if f and g are both functions that are continuous at some point a, then the difference f - g…

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Q: Prove that the identity function is differentiable at every x E R

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Q: Determine whether the function f (x) = {: —х in — 1 <x<0 in 0<x < 1 is even or odd,

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Q: Prove that f(x) = x/k can be defined on (0, co) by the require- ment that it be the inverse function…

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Q: Consider the function f(x)= 4x+cosx (a) Show that f(x) is one-to-one.  (b) Compute   (f -1)´(2π)

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Q: The domain of the function f(x)= cosh"'(x/ 3) is Select one: O[3,) O[0,3] O None O[-3,3 ] OR-{3}…

A: The domain of the function f(x): The set of values of x for which the function f(x) is defined.

Q: Derive a numerical differentiation formula of order O(h4) by applying Richard- 4. extrapolation to…

A: Its given that:

Q: Suppose that f(x) is continuous and differentiable, f(2) = -2, f(6) = -6, and f(x)dx =-10. Compute ,…

A: Topic:- Functions

Q: Let g be a function that is defined for all x, x # 2, such that g(3)% = 4 and the derivative of g is…

A: To find: Critical points of g where g'(x)=x2-16x-2,x≠2.

Q: Q2: Is the function f(x) = sin x -2x + x' odd or even Odd function C Even function C Not of all O

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Q: Show that if f is symmetric with respect to x = a, then g(x) = f (x + a) is even.

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Q: Let f (x) = 3x+ – 10x – 12x² + 12x – 7. On what interval is this function concave | downward? O (-0,…

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Q: The function of f is increasing on I if ______ when x1 < x2.

A: We have to find when function is increasing

Q: Let f' : N → R be a function defined as f'(x) = 4x2+ 12x + 15. Show thatf : N→ S, where, S is the…

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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.