. If x, y E R with x < y, prove that x < tx + (1 – t)y < y for all t e (0,1).

Q: prove if x → y then xor y = y =

A: To prove : If x→y then x or y = y Lets use the help of truth table to prove the above statement.

Q: is continuous at x=3, f (-1) =6 ? + a, x # 3 3b, x 3 ontinuous at x=2? - x > 0

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Q: Let R={(x,y) ERxR: |x| = |v|). Then R a function on R. оо True False

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Q: 3. Find the domain of the following functions: a.) f(x) = V1-x Inx b.) f(x) = Vx - 1lr

A: solutionsince the given (3)(c)f(x)=1-xlnx(b) f(x)=x-l ln(x)use have to find the domain of the…

Q: If y =f (x) is a function in x then prove that f(a + nh) ="C,f(a) +"C, Aƒ(a) +"C,ƒ(@) + +"C, A"…

A: Given: y=fx is a function in x Prove that: f(a+nh)=C0nfa+C1n∆fa+C2n∆2fa+…+Cnn∆nfa…

Q: 5. Let f be defined for all real numbers and suppose f(x)-f(y)| ≤ (x - y)² for all real numbers, x…

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Q: e equation of

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Q: 3. Let f(x, y) = (2x – y,x- 2y); (x, y) E R x R, (R is set of real numbers.) a) Show that f is one…

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Q: Q.3) Find the constants a, b, c, d, e if f(z) = (ax³ + bxy² + 3x² + cy² + x) + i(dx²y - 2y³ +exy +…

A: Given :  f(z) = ax3+bxy2+3x2+cy2+x + idx2y-2y3+exy+y  is analytic. To find : The constants a, b, c,…

Q: Determine the length of x = 3+ (y – 2)², 4<x< 19

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Q: Which graph corresponds to the function given? h(x) = {x², x 1

A: Here we have to draw the graph of the function.

Q: How many functions f:R→R  exists such that : 2f(x)+2f(y)=f(x+y)+f(x−y)+2y holds for all real x and…

A: By choosing x=y=0. Also by choosing x=y we can see f(2x)=4f(x).  Moreover choose x=0 and use f(0)=0,…

Q: 3. If f:(X,r)(Y.A) and g:(Y,A)→(Z,r) are open functions, prove that gof :(x.r)(Z.r) is open…

A: Definition: An open map is a function between two topological spaces that maps open sets to open…

Q: 6. Prove that |xy| = |x|ly| for all real numbers x, y.

A: The given problem is to prove the following statement. Given statement is |xy|=|x||y| for all real…

Q: if f (r) is differentiable prove that f(r)r is irrotational

A: The objective here is to prove that the vector field f(r)r is irrotational. Given: f(r) is…

Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

A:

Q: 1. Let f(x,y)=x²y + √y Find each value a.f(2,1).b.f(3,0).c.f(1,4)

A: We have to find the value of the function for the given points, f(x,y) = x2y + y points : f(2,1)…

Q: 5. Show that the functions x, x² interval [-1,1]. x are mutually orthogonal in the -

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Q: Let R be a commutative ring and let A be an ideal of R. Show that VA = {x € R:x" e A for some…

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Q: Suppose that Y = Z - X is independent of Z and of X. Show that Y is a constant.

A: Given that Y=Z-X. To show: Y is a constant function.

Q: 5. Show that the functions x,x? -,x³ -x x are mutually orthogonal in the interval [-1,1].

A:

Q: Determine the value of m such that the function x- y - 5 E + 5 VI - Vỹ f(x, y) = is continuous at…

A:

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: 2. Let f (a, b)→ R be a function and suppose there exists constants M > 0 and a > 1 such that |f(x)…

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Q: be the vector spaces consisting of zero and polynomials in z of degree <3 and degree < 6…

A: Given that  and  are two vector spaces consisting of  and polynomials of degree  and degree…