(b) Let f be a Cn function on an open interval [a, b]. Assume that the set of zeros {x € (a, b) : f(x) = 0} has at least (n + 1) elements. Prove that there exists xo E (a, b), such that f(m) (xo) = 0. Use (b) to prove that a polynomial (with real coefficients) of degree n has at most n real roots.

Q: Let f(x) be a polynomial function whose coefficients areintegers. Suppose that r is a real zero of f…

A: Consider an example:  2x3=x+6x+12x-42x3=x3+14x2-2882x3-x3-14x2+288=0⇒x3-14x2+288=0

Q: 1 D 1 4 -1 -2 (b) Your friend attempted part (a) and obtained a negative value. "That doesn't make…

A: As per the question we have to find the double integral analytically and deduce that the value is…

Q: Let f(r) = (8+2)r Find the third Maclaurin polynomial for f. Use the polynomial in (a) to evaluate…

A:

Q: Consider the polynomial equation of P(r) = 0 where P(x) = #* – a* – ² – a – - 2. - T (a) What are…

A: The given polynomial is Px=x4-x3-x2-x-2.

Q: If f(x) is a polynomial of degree less than 3, then prove that f(a) (x-a) |1 f(b) (x-b) f(c) (x-c) 1…

A:

Q: Draw a function f(x), defined for all real numbers x, with one local maximum and two local minimum.

A:

Q: Viewed as a function f: R → R, a polynomial of odd degree is always surjective. True False

A:

Q: Let f(z) be an entire function satisfying |ƒ (²1 +22)| ≤ |ƒ (2₁)| + + + |ƒ (2₂) | = n for all 2₁, 22…

A: As per the question we are given an entire function which satisfies the following inequality :And…

Q: According to the integral zero theorem, the possible integral zeros of f(x) = 4x³+5x²+7x+6 are…

A:

Q: Let f and g be the functions defined by f : R → R+ ∪ {0} with f (x) = x2 and g : R+ ∪ {0} → R with…

A: (fog)(x) is the value of the function f(x) at g(x) (fog)(x)= f(g(x))

Q: Determine the canonical form of the function f : R → R, f(x) = x2 − 4x + 6 and sketch its graph.

A: To determine the canonical form of f(x)=x2-4x+6 and to sketch its graph

Q: Draw a function f(z), defined for all real numbers z, with two local maximum and one local minimum.

A: We will draw a function which has two local maximum and one local minimum.

Q: let f(z) be polynomial of degree at least two. show that |f(z)|>|z| for large |Z|.

A: We prove this result by contradiction.

Q: Find Ao, A1, and Az in rewriting the function of f(x=2vx 0マx in terms of Lagure polynomial ョーの

A: Given: Let us consider the given function  fx=2x       x≥00             x<0 and the Lagure…

Q: Viewed as a function f: R → R, a polynomial of even degree is never injective. True False

A:

Q: Find the integers that are upper and lower bounds for the real zeros of the polynomial (show all…

A: The objective is to find the integers that are upper and lower bounds for the real zeroes of the…

Q: Let p: R → R be a polynomial of odd degrees. Prove that the equation p(x) = 0 has a solution in real…

A: Given p:R→R be a polynomial of odd degree. We have to prove that the equation px=0 has a solution in…

Q: Show that f(x) = e describes a bijective function from R to (0,+x). (a) Show that f is a function.…

A:

Q: Find functions f and g from the set of positive integers to the set of real numbers such that f (n)…

A: Here, we need to find functions f and g from the set of positive integers to the set of real numbers…

Q: Prove that you can find a polynomial p(x) = Ax2 + Bx + C that passes through any three points (x1,…

A: we have to prove that we can always find a polynomial p(x) = Ax2 + Bx + C that  passes through any…

Q: Suppose that a function ƒ(x) has the property that ƒ(x1 x2) = ƒ(x1) + ƒ(x2) for any two positive…

A:

Q: V5 + v6 Find a monic integral polynomial for which x is a root and prove that x is algebraically…

A: We have x = 5+6. The objective here is to show that x is an algebraically irrational number.  An…

Q: Find an 4th degree polynomial function with real coefficients whose zeros are 4,–1,2, v2i and a…

A: Here degree of polynomial is 4 so it has 4 roots.