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

Q: 1. Explain why both f(x)= x³+ 1 and g(x) tions. x3+1 are rational func-

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Q: The function f(x) = 2x° - 36x + 192x 8 has two critical numbers. %3D The smaller one is x = and the…

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Q: b.) Find the root of f(x) = 2x³-5x+2 using Method of successive substitution. (Allowable error <…

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Q: Find the values of a and b such that f(x) = x³ + ax² + bx + 5 has x = 1 and x = 2 as critical…

A: The function f(x)=x3+ax2+bx+5 has critical points at x=1 and x=2

Q: 1. If the leading coefficient of an odd-degree polynomial function is negative, then the function…

A: Check the end behaviour of the function as x tending to minus infinity and infinity.

Q: The function f(x) 2x 36x + 162I - 7 has two critical numbers. The smaller one is x = and the larger…

A: The critical number of a function f is a number c such that either f'(c) =0 or f'(c) doesn't exist.

Q: Find the critical numbers of f(x) = ln(x2 + 5x + 14).

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Q: Let g (x, y) = 9 sin(xy) – 11x² In(y) + 4. Find the degree 2 polynomial, p, which best approximates…

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Q: The function f(x) = 2x³ - 39x² + 216x 3 has two critical numbers. The smaller one is x = and the…

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Q: 2. If the polynomial f (x) divided by (x – 3) gives a remainder of -1, then the remainder of…

A: Solution is given below:

Q: The function f(x) 2x – 33x2 + 108x – 1 has two critical numbers. The smaller one is x = and the…

A: Explained below

Q: Find a cubic polynomial with a critical point at x = 2, and inflection point at (1,4), and a leading…

A: Find a cubic polynomial with a critical point at x = 2, and inflection point at (1,4), and a leading…

Q: 20. The function e* outgrows any polynomial faster as x→ oo than any polynomial Show that e grows…

A: Here we have to show that exponential function grows faster than any polynomial as x goes to…

Q: Let f(x) = x³ + x + 4 and g(x) = 3x + 2 in Zs[x]. Then, upon dividing %3D f(x) by g(x), we get None…

A: f(x)=x3+x+4 and g(x)=3x+2 in Z5[x].

Q: Find the location of the global maximum and minimum of f(x) = x^3-6x^2+1f(x)=x3−6x2+1 on the…

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Q: Let f(x) = x' + 2x + 4 and g(x) - 3x + 2 in Z,[x]. Then, upon dividing f(x) by glx), we get 7- The…

A: I am solving part7 &amp; part8 as you mentioned.

Q: Find the absolute maximum and minimum values of f(x)= ln(x^2+3x+15) on the interval [-2,1]

A: We find f'(x) using chain rule

Q: The 2nd degree polynomial that interpolates to f(x): = 2cosx - sinx at x = 0, x = π/2 and x = π is…

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Q: Find the critical numbers of the function f(x) =x^6/5-12x^1/2

A: In the given question find the critical point of the given function: f(x) = x^(6/5) - 12 x^(1/2)

Q: Let f(x) = x3+ x +4 and g(x) = 2x + 3 in Z3[x]. Then, upon dividing f(x) by g(x), we get The…

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Q: If f(x)=ln(x) and x,-3.1, x₁-3.2, x-3.16, then the error bound in Lagrange interpolating polynomial…

A: Given the function is fx=lnx and x0=3.1, x1=3.2, x=3.16

Q: Let f(x) = x³ +x + 4 and g(x) = 3x + 2 in Zs[x). Then, upon dividing f(x) by g(x), we get None of…

A: Every integer polynomial we can write as an polynomial in Z5(x). Using this idea again and again we…

Q: 3. f(x) = x2- 4x- 5 4. f(x) = X2 - 9x - 36 (x-12)(x+3) roots are x=12 x=-3 5. f(x) = x2+ 3x- 40 6.…

A: Given3)f(x)=x2-4x-55)f(x)=x2+3x-406)f(x)=x2+22x+21 To find Roots of above

Q: Find an nth degree polynomial function with real coefficients satisfying the given conditions. n =…

A: The objective is to find an nth degree polynomial function with real coefficients satisfying the…

Q: (a) If f(x) = 4x³ + 15x² – 72x + 5, then find 1) The critical numbers of the function f(x); 2) The…

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Q: Express the polynomial x3 + 2x + 3 EZŞ[x]as a product of irreducible polynomials of Z5[x].

A: Explanation of the answer is as follows

Q: Find the critical numbers of the function.…

A: Critical numbers : f(x)=4x2-6x The derivative of the function is f'(x)

Q: The average pollen count in a certain city on day x of the pollen season is P(x)=8x − 0.2x2 (for 0…

A: Consider the given function,  Represent the pollen count in a city on the day.   The day, which the…

Q: OFind a root of an equation f(x)=2x-2x-5 using Newtor aphson Method

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Q: Find all the critical numbers for the function g(x) = ¼ x4 - x3 - 5x2

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Q: Find Absolute Extremums of F(x)=2x^3-6x over [0,3]

A: To find the absolute extremums of the function F(x)=2x3-6x over [0,3]

Q: The function f(x) = (6x + 3)e-6a has one critical number. Find it.

A: Given: f(x) = 6x + 3 e-6x To find: Critical number  Proof: We will differentiate the function w.r.t…

Q: Using Taylor's theorem, formally represent 2x° +x* – 3x³ +x² – 5x +1 as x+1.

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Q: The Taylor polynomial of degree 100 for the function ƒ about x = 3 is given by (x – 3)100 50! 2n…

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Q: In the function:  f(x)= (7x+B)/(4x+b) , with 7b&gt;4B What are the critical numbers of the function?

A: Given:

Q: Use the second Taylor polynomial P2(x)P2(x) for f(x)=x2+ln(1+x)f(x)=x2+ln⁡(1+x) about x0=0x0=0 and…

A: P2x=f0+f'0x-0+f''0x-02f(x)=x2+ln(1+x)f(0)=0f'(x)=2x+11+xf'(0)=1f"(x)=2-1(1+x)2f"(0)=2-1=1∴P2(x)=x+x2…

Q: g(x) = x' + 3 x² – 24 x

A: The point x=k is said to be the critical point of the function f(x) if Case 1 : f(k) exists and…

Q: f(x) = x3 + 5x² + 2x

A: Given: f(x)=x3+5x2+2x

Q: The polynomial f (x) leaves the remainder x + 2 at the division with x ^ 2 + 2x-3 and leaves the…

A: Use the remainder theorem for the polynomials to determine the remainder when f(x) is divided by…

Q: Sketchthegraphofy=x2−5x+4.

A: We first make a table for y=x2 −5x+4 x 1 2 2.5 3 4 y 0 -2 -2.25 -2 0

Q: If the leading coefficient of an odd-degree polynomial function is positive, then the function…

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Q: Find all critical numbers of the function g(x) = x^4 - 10x^2

A: Using definition of critical numbers.

Q: if xo = 2, x1 = 3 and x2 = 4 the second degree %3| polynomial for f(x) = using Lagrange…

A: Given function is f(x)=12and the nodes are x0=2, x1=3 and x2=4. We have to find second degree…

Q: Find the critical number of f(x) = 3x^2 - 18x + 4

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Q: Let g (x, y) ) = 11 sin(xy) — 13x² ln(y) + 20. Find the degree 2 polynomial, p, which best…

A: Given: The function is g(x,y)=11sin(xy)-13x2ln(y)+20 To find: The degree 2 polynomial which best…

Q: Find the critical numbers of the function. f(x)=32x-2x^2 x=

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Q: 1. If the rational function y = r(x) has the vertical asymptote %3D x = 2, then as xr→2*, either…

A: Given: y=rx

Q: The 2nd degree polynomial that interpolates to f(x) = cosx + sinx at x = 0, x= π/2 and x = π is best…

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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.