Let f(x) = x³ + x + 4 and g(x) = 3x + 2 in Zs[x]. Then, upon dividing f(x) by g(x), we get The quotient is 6 +x + 4x^2 and the remainder is 0 The quotient is 4+2x+2x^2 and the remainder is 1 The quotient is 1+3x + 3x^2 and the remainder is 1 O None of these

Q: Find all the rational zeros of the function f(x) = 2x - 9x4 + 9x³ - 3x² + 7x + 6. -1, 2 x = 을 3,2 x…

A: Roots satisfyf(x)=0f(x)=2x5-9x4+9x3-3x2+7x+6f(-1)=-24f(1)=12Options, A and E ruled…

Q: If f(x) = x* + 7, g(x) = x - 5, h(x) = Va, then fo g(x)= go f(x)= ho g(x)= %3D

A:

Q: Verify that the factorization for f (x) = x3 + x2 + 1 over Z2

A: Given that, fx=x3+x2+1. Also ℤ2=0, 1 To verify the factorization for fx over ℤ2.

Q: What is the quotient when f(x) = 6x¹ − 2x³ + x² − 3x + 1 is divided by g(x) = x² + x -2 in Z₂[x]?

A:

Q: d(x) = x - 1, f(x) = x²00 factor not a factor

A:

Q: Find the remainder when f(x) = x7 + 3x5 + x - 1 is divided by x2 +1 in Z5[x].

A:

Q: If f (x) = 3x ^ 4 + 5x ^ 3 + x ^ 2-2 then write f (x) as the product of irreducible terms of Q

A: We have to express f(x) as the product of irreducible terms of Q.

Q: The domain of the function f(x)=√12+4x−x^2 is the closed interval [A,B] where A= and B=

A: Domain is the set of all values that can be taken by x so that the equation is defined.

Q: Differentiate. f(x) = (x 2 - 2x + 4)ex O ex(x² + 2) O ex(x² + 4x + 2) 3 +x+4 O ex(2x - 2)

A: Given the function fx=x2-2x+4ex

Q: Let f(x)=√(x+1/x),g(x)=2/√(x-1) and h(x)=x^2+1 Describe the domains of f(x) and g(x) . Find…

A: We have to find domain

Q: Given f(x)= x^3 − 6x^2+9x, find the multiplicity of the zero at x = 3

A:

Q: h = (-6x³ + 9xª + 5x7) (7 + 5x – 4x?). Then h= fgwhere %3D f = and g = Using the product rule h'(x)…

A: differentiate this with product rule.

Q: Show that the product rule works for the case f (x) = x2, g (x) = x3.

A: Given that, fx=x2gx=x3 We have to show that the product rule works for the case of given functions.

Q: G(7/2)=2(7/2)^2+6 function of g(x)=2(x)^2+6 where G(x) is 7/

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Q: a) Suppose that people arrive at a bus stop in accordance with a Poisson process with rate 2. The…

A: Given that : The amount of waiting time= X Number of arrivals at time t = N(t) PART A The value of :…

Q: B. Let f(x) = x – x - 4 and g(x) = 2x – 6 4.) Find ()(x - 1) 5.) Find (4) (x)

A: topic - functions

Q: Consider f(x) = 8x +3, g(x) = x³. fog= 2x^3+3 %3D go f = (2x+3)^3 f(x) = Vx² + 5, g(x) = x² + 6.…

A:

Q: Let f ( x ) = x^ 2 − 1 and g ( x ) = 3 x. Find ( f ∘ g ) ( x ). group of answers 3x^3 − 3x 3 x^2…

A: Given functions, f(x)=x2-1 and  g(x)=3x

Q: Find the hcf of f(x) = x^5 + 3x^4 + 10x^3 + 19x^2 + 22x + 32 and g(x) = x^3 + 3x^2 + 7x + 10 in…

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.