Question

Show that if a, b, c are real numbers such that b > (1/3)a^2 , then the cubic equation x^3 + ax^2 + bx + c = 0 has precisely one real root

Answer #1

Let
a, b and c be real numbers with a does not equal to 0 and
b^2<4ac. Show that the two roots ax^2+ bx + c =0 are complex
conjugates of each other.

Let f(x) be a cubic polynomial of the form x^3 +ax^2 +bx+c with
real coefficients.
1. Deduce that either f(x) factors in R[x] as the product of
three degree-one
polynomials, or f(x) factors in R[x] as the product of a
degree-one
polynomial and an irreducible degree-two polynomial.
2.Deduce that either f(x) has three real roots (counting
multiplicities) or
f(x) has one real root and two non-real (complex) roots that are
complex
conjugates of each other.

15.)
a) Show that the real numbers between 0 and 1 have the same
cardinality as the real numbers between 0 and pi/2. (Hint: Find a
simple bijection from one set to the other.)
b) Show that the real numbers between 0 and pi/2 have the same
cardinality as all nonnegative real numbers. (Hint: What is a
function whose graph goes from 0 to positive infinity as x goes
from 0 to pi/2?)
c) Use parts a and b to...

Exercise 1. For any RV X and real numbers a,b ∈ R, show
that
E[aX +b]=aE[X]+b. (1)
Exercise2. LetX beaRV.ShowthatVar(X)=E[X2]−E[X]2.
Exercise 3 (Bernoulli RV). A RV X is a Bernoulli variable with
(success) probability p ∈ [0,1] if it takes value 1 with
probability p and 0 with probability 1−p. In this case we write X ∼
Bernoulli(p). ShowthatE(X)=p andVar(X)=p(1−p).

1) find a cubic polynomial with only one root
f(x)=ax^3+bx^2+cx +d such that it had a two cycle using Newton’s
method where N(0)=2 and N(2)=0
2) the function G(x)=x^2+k for k>0 must ha e a two cycle
for Newton’s method (why)? Find the two cycle

Let p and q be two real numbers with p > 0. Show that the
equation x^3 + px +q= 0 has exactly one real solution.
(Hint: Show that f'(x) is not 0 for any real x and then use
Rolle's theorem to prove the statement by contradiction)

Continuity and the derivative:
1A) Show that there exists a real root of the equation in this
interval: cos(root x) = e^x-2 [0.1]
1B) If f(x) is a continuous function (on the reals) that has
only one root at x=2, and if f(4)>0, can f(3)<0? Explain.

Show that the equation
x+sin(x/3)−8=0
has exactly one real root. Justify your answer.

A) Show that there exists a real root of the equation in the
given interval cos (roots)=e^x-2 [0.1]
B) For the piecewise function, calculate the unknown values that
allow the functions to be continuous everywhere.
- g(x)=
1. Ax-B, where x is less than or equal to -1
2. 2x^2+3Ax+B, where x is bigger than -1, and less than or equal
to 1
3. 4, where x is bigger than 1

Suppose that f(x)=ax^3+bx^2+cx+d cubic polynomial.. Show that
f(x) and k(x)=f(x-2) have the same number of roots.(without
quadratic formula)

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 19 minutes ago

asked 24 minutes ago

asked 24 minutes ago

asked 42 minutes ago

asked 42 minutes ago

asked 55 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago