. Let f (x) = 11x +9x +21. Prove that f (x) is irreducible over Q.

Q: Is f(x)=2x+4 reducible over Z ニ

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Q: Z→ Z be such that f(x) = 4x + 2. Is f invertible, and if so, find its inverse function.

A: Given query is to find the inverse function.

Q: Show that : f(x) = x2 + x + 2 e Zg[x} is irreducible. Further, find the order of f (x).

A: Definition: 1. A polynomial f(x) ∈ ℤp[x] is said to be irreducible if and only if it has no root in…

Q: The f(x) 6x2 -5x-4 2x2 +x is discontinuous %3D at x = -1/2. By redefine = w , then f will be…

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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: Let g=yf(xy). Show that: - дд у ду дд - Х = g дх -

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Q: Define f: [0,2]-R by f(x) = x²+2x-1 , then the sup f- inf f is : [0,2] [0,2] 1.a O V8 b 0 3.cO 8 dO

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Q: Let f, g: R → R defined by if x 1 f(x) = %3D and g(x) = 3x + 2. %3D fog and go f.

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Q: For what values of a and b is ax + 2b, g(x) = {x + 3a - b, 0 2 continuous at every x?

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Q: ) If f(x) = (0 <* < 1) (-1 < * < 0) expand f(x) in the form c,P(x).

A: According to our company policy i am supposed to answer ony one question at a time.  For the rest…

Q: Let f : Z → Z be such that f (x) = x + 1. Is f invertible, and if it is, what is its inverse?

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Q: 2. Find the linearization L(x) of f(x) = x³ - 2x + 3 at x = 2 and use it to approximate f(1.9) and…

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Q: Let f(x) = x²+x+ 2 = Z₂ (x) prove that f(x) irreduciable Poly

A: The given function is  f(x)=x2+x+2.

Q: Let f, g:R R defined by if x 1 (x2 and g(x) = 3x + 2. %3D Find f-1 and g-1.

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Q: Show that the function f : N → N, given by f(x) = 2x, is one-one but not onto.

A: If f(x1) ≠ f(x2) then we get:

Q: Let z = f(x, y) = 4x² + 2xy + By, Step 1 of 4: Evaluate f(3, 1).

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Q: Determine b so that f(x) is continuous if O 8 x + 6 xS 6 f( x ) = D O 2 x2 + bx +10 x > 6 b =

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Q: Let f(x) = x3 + 6 ∈ Z7[x]. Write f(x) as a product of irreduciblepolynomials over Z7.

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Q: (d) If f(x) is irreducible over Q then f(x+ a) : a e Z is irreducible over Q.

A: Use the property of reducible to start the proof.  According to the property of contrapositive, "if…

Q: b- For the following functions prove that ax2 ay2 1- f(x,y) = e-2y cos 2x %3D 2- f(x,y) = Inx2 + y2…

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Q: Show that f(x1, x2) = 2 - (x1 - 1)^2 - e^x2^2 is concave.

A: the given function is fx1,x2=2-x1-12-ex22 To find the function is concave we have to show all its…

Q: Let g=yf(xy). Show that: - მყ У ӘУ дд - X- = g дх

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Q: Show that the following polynomials are irreducible over Z[r]. fi(x) = 17r³ + 7x + 3, f2(r) = 2r* +…

A: Using the result: A polynomial of degree 2 or 3 is reducible over the field F if and only if it has…

Q: For what value of a is x² -1, x< 3 f(x) 2ax, x2 3 continuous at every x?

A: Given function is Since, f(x) is continuous at every x, so f(x) is continuous at x=3.

Q: The function f(x) =x + 1/x is concave upward on none O (-1) O (0,) O (0,1) O (1,)

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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: Prove, if f(x)= V1+x² –In +1+ 1 is increasing and concave on x X E (0;+∞) .

A: f is increasing if first derivative is positive in the interval  and f has concave if second…

Q: 3. Let f (x) = x ² + 8x + 14. Find all the values interral C-G ,2) such that f (c) = 0 of ċ in the

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Q: . Suppose that f is a polynomial of degree 1 and s1, is the linear spline that interpolates f at the…

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Q: b) Show that the function f(x) = x7 +1 is invertible %3D

A: Given that  f(x)=x^7+1

Q: Suppose that f (0) = 2 and f' (x) ≤ 3 for x &gt; 0. Apply the MVT to the interval [0, 4] to prove…

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Q: (c) Show that : f(x) = x2 + x + 2 e Zg[x] %3D is irreducible. Further, find the order of f (2).

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Q: Show that the function f(x) = x’ + 1 is invertible from R to R.

A: First prove that the  function is one -one  let f(x)=fyx7+1=y7+1x7=y7x=y therefore, the function is…

Q: Find the value of c which satisfies the conclusion of Rolle's Theorem with the given f (x) =x³ - 3x,…

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Q: Prove the following relations: ag ag (x - t) at Əx Where g(x,t) is the generating function of…

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Q: Showthat fo= x| is Continuous evalyuhore- 1x1+1

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Q: - Let Q(x) be a polynomial of degree less than N. Prove that Q(x)PN(x)dx = 0.

A: Note, that this result is a specific case of orthogonality of Legendre’ s polynomials. We will prove…

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: Let f(x) = cos(E) and P(x) be the interpolation polynomial of degree at most 3 which agrees with…

A: Neville’s method generates Lagrange’s approximations recursively. In Neville’s method, we denote…

Q: Find m so that f(x) is continuous at x = 2. x +4 if x 2 -

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Q: Find the Wronskian of the given pair of functions: x, xe* O xe* O x O et

A: The solution is given as

Q: Q3. Prove that if g interpolates the function f at ro, T1, . .. , In-1 and if h interpolates f at…

A: We have to prove that if g interpolates the function f at x0,x1,…,xn-1 and if h interpolates f at…

Q: Let S«) = {r²+2x}°, 121, ax+bx, x< 1. Compute for all a and b such that the function f(x) is…

A: We have to find

Q: Suppose that only the points (x,y), j = 1 ... 16, that lie on the graph y = f(x) of a function f(x)…

A: The interpolation techniques are used to establish the functional relationship between any two…

Q: inf f. is not measurable Vn. i

A: False ; is sequence of measurable functions fn is measurable then inf fn is measurable .

Q: Answer all the following questions 1. Find the doma in of f(x) = x-4

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Q: 2. Find the principal part with respect to x for x 0 of the infinitesimal f(x) = = Vx + Vx. (That…

A: We have to use expansion of the function after some manipulations.

Q: Find a finite difference approximation of Cf(x) + Df'(x) using the points f(x – h), f(x), and f(x +…

A: Let f be any function defined on a vicinity of the fixed x , and let h &gt; 0 representing apositive…