Show that f(x) = x4 + 4.x + c has at most two real roots for any umber c.

Q: B) Estimate the next root of the function f(x)=x^3-x-1 using Muller method, let x0=1, x1=2, x2=1.5

A: Muller method: The Muller approach is applicable to any root-finding programme, but it is especially…

Q: Let g (x, y) = 7 sin(xy) – 9x? In(y) + 4. Find the degree 2 polynomial, p, which best approximates g…

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Q: (a) Find the quadratic approximation P2 for f (x) = In x at x = 2 (b) Use above to approximate…

A:

Q: 3. Compute the value of F.dr if F = -yi + xj and C is a semicircle from (-1,1) to (1,-1).

A:

Q: Find the third Tayler polynomial P, (x) f fxl= cos X at

A: Given function is  fx=cosxat x=π

Q: Show that f(r) = O(|x – xo|*) and g(x) = o(]x – roP) imply f. g(x) = o(|x – xo]k+i). - -

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Q: 4. Use the error formula to find a bound for the error at a = 0.31 in approximating a sin(2x-5+1 on…

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Q: Let f(x)= x2 − 6. Prove that the root of this function is equal to √6 {correct to 2 dp}, and lays…

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Q: If p(x) = p(t)dt, prove p must be the zero function. Hint: Fundamental Theorem of Calculus.

A: Fundamental theorem of calculus. Let f be a continuous function on [a,b] . If Fx=∫axftdt   then…

Q: Form the interpolating polynomial of degree k that satisfies pG) (xo) = f(G) (xo) for 0<j< k.

A: To form: The interpolating polynomial of degree k that satisfies pjx0=fjx0 for 0≤j≤k

Q: Let x, y be real numbers such that xy=1x and x≠y. Prove that x^2+y^2≥2\sqrt{2}|x−y|.

A: Given: xy=1x,x≠y To find: x2+y2≥22x-y

Q: Use the Second Fundamental Theorem of Calculus to find F'(x).

A: The function is given as

Q: Prove that f(x) = [[sin²(2log r)– 3 sin ( 2logt)+2] dt %D 1. Cannot be decreasing any where,

A: To prove that f(x) can not be decreasing anywhere, where f(x) is defined as…

Q: 3. Use Müller's Method to determine all the roots of f(x)=x' -x² -x-2 Choose x2=1.0, xo=X2+Ôx2,…

A: we have, fx=x3-x2-x-2 x0=1.01x1=0.99x2=1 1st iteration:…

Q: Ex.5 - Prove that the polynomial 2ª + z + 1 has exactly two zeros in the right half-plane.

A: Solution: We will be using the method of Routh array For a polynomial Qz=anzn+an-1zn-1+...+a1s+a0…

Q: Show that the nth Maclaurin polynomial for f(x) = e* is Tn (x) = 1+ 2! n!

A:

Q: 7. Suppose f(x) = x³ – 27x + 1. Prove that f has exactly one root (zero) in the interval [0, 2].

A:

Q: Use the intermediate value theorem to show that f(x) = x^4 + 8x^3 −x^2 + 2 has a zero in I = [−1,…

A: Given,

Q: Find the Maclaurin polynomial approximation of degree n for the function f(x)= e2x , and write a…

A: Maclaurian polynomial: The Maclaurin polynomial approximation of degree n for  the function f(x) is…

Q: Obtain a quadratic polynomial approximation to f (x)= 3e* using Newton divided difference…

A: Given:        f(x)=3ex       Here h=b-aN                  =0.5-04=0.125     The value of table for x…

Q: Show that f (x) = 1 − |x| satisfies the conclusion of the MVT on [a, b] if both a and b are positive…

A:

Q: - R-{-}→R-{} defined by f(x) = Show that f: R 3x-5 2x+1 is bijective.

A: Given function is f: ℝ - -12 → ℝ--32 such that f(x) = 3x-52x+1

Q: Use the Intermediate Value Theorem to show that there is a root of the given equation in the…

A: The given equation is: lnx=x-x, 2,3 Use the intermediate value theorem to show that there is a root…

Q: use lagrange multipliars to maxim The Fungtien 7CX>y)=V57-x" -y subjeet to the Following canstaint…

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Q: Apply Newton's forward formula to obtain a third-degree polynomial that interpolates y = 2 cos(nx)…

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Q: Find the Taylor polynomial Tz(x) of f(x) = In x at x=

A: Given:

Q: Show that the function f(x)=3x-2 is a bijection.

A: For a function is to be bijective, then the function should be well defined, surjective, and…

Q: Suppose that K, L, M, and N are four approximations of the unknown root r of function f such that…

A: Given: Suppose K,L,M,N are four approximations of the unknown root r of function f. since r is the…

Q: sin

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Q: List all the polynomials of degree 2 in Z2[x]. Which of these areequal as functions from Z2 to Z2?

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Q: Verify that the hypotheses of Rolle’s Theorem are satisfied for f(x)=x^2-14x+45 on the interval…

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Q: Given that f(x) = 2x + cos(x) is a bijective function, find (f-')'(1)

A: The function f-1 represents the inverse of function f. Therefore, if f-1x=y then x=fy represents the…

Q: By using Ho(x)=1 and Hi (X)=2x in Hermit Polynomials, caleulate the Valne of Hg L)? (1)

A:

Q: find the absolute maxima and minima of the func-tions on the given domains. ƒ(x, y) = x2 + y2 on the…

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Q: Find a cubic polynomial f(x)=ax^3+bx^2+cx+d that has horizontal tangents at the points (-3,2) and…

A: Given f(x)= ax^3+bx^2+cx+d that has the horizontal tangents at points (-3,2) and (2,-4). We need to…

Q: Use the Cauchy-Riemann Equations to show that f(z) = z is differentiable for 16. all z and find…

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Q: Suppose that K, L, M, and N are four approximations of the unknown root r of function f such that…

A: Given: f(K) = 0.001 f(L) = -0.11 f(M) = 0.0012 f(N) = -0.011 are the roots values of the unknown…

Q: Verify that f(x)=x2+1/x satisfies the hypotheses of the Rolle's Theorem on the interval [1/2 2] and…

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Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

A: The given function is f:ℝ→ℝ defined by fx=5x-3. We say that a function f is bijective if it is both…

Q: Let p(x) be a polynomial function of degree n with simple roots r1,..., xn. Show that ) p"(x;) = 0.…

A: Let Pn be a given statement for all natural numbers n. If  i. P1 is true ii. Pk+1 is true whenever…

Q: 10. If f(x) = x³ – 3x +1 is defined on [0,1], then by using False position method, find the first…

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Q: The domain of f(x) = /1 – I In(x) is %3D

A: The domain is the set of all possible x-values which will make the function give real y-values as…

Q: Letf(x) be a polynomial with integral coefficients. Iff(1) and f(2) both are odd integers, prove…

A: Let us assume that  f(x)=0 for some integer x=k, then x-k divides f(x). therefore  f(x)=(x-k)g(x) ;…

Q: Prove that if the (n + 1) th derivative of a polynomial vanishes but the nth does not, then the…

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Q: The domain of the finction f(x, y) = In(-x² - +1) is:

A: The domain of a function is the range of the values that can be input into the function. The…

Q: Use the IVT to find an interval of length 0.3 that contains a root of In r = 3 – 2r.

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Q: Find the linearization L(x) of f(x) at x = a. f(x) = 3square root of x, a = 27 show all work

A: fx=3x, a=27 Differentiating with respect to x, we get f/x=3·12x    ∵dxdx=12x=32x Now,…

Q: Use the error bound for SN to obtain another proof that Simpson’s Rule is exact for all cubic…

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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.