(a) Use Newton's method to find the critical numbers of the function
f(x) = x6 ? x4 + 2x3 ? 3x
correct to six decimal places. (Enter your answers as a comma-separated list.)
x =
(b) Find the absolute minimum value of f correct to four decimal places.

Answers

Answer 1

The critical numbers of the function f(x) = x⁶ - x⁴ + 2x³ - 3x.

x₅ = 1.35240 is correct to six decimal places.

Use Newton's method to find the critical numbers of the function

Newton's method

[tex]x_{x+1} = x_n - \frac{x_n^6-(x_n)^4+2(x_n)^3-3x}{6(x_n)^5-4(x_n)^3+6(x_n)-3}[/tex]

f(x) = x⁶ - x⁴ + 2x³ - 3x

f'(x) = 6x⁵ - 4x³ + 6x² - 3

Now plug n = 1 in equation

[tex]x_{1+1} = x_n -\frac{x^6-x^4+2x^3=3x}{6x^5-4x^3+6x^2-3} = \frac{6}{5}[/tex]

Now, when x₂ = 6/5, x₃ = 1.1437

When, x₃ = 1.1437, x₄ = 1.135 and when x₄ = 1.1437 then x₅ = 1.35240.

x₅ = 1.35240 is correct to six decimal places.

Therefore, x₅ = 1.35240 is correct to six decimal places.

Learn more about critical numbers here:

brainly.com/question/29743892

#SPJ4


Related Questions

Find the general solution to the following differential equations:
16y''-8y'+y=0
y"+y'-2y=0
y"+y'-2y = x^2

Answers

The general solution of the given differential equations are:

y = c₁e^(x/4) + c₂xe^(x/4) (for 16y''-8y'+y=0)

y = c₁e^x + c₂e^(-2x) (for y"+y'-2y=0)

y = c₁e^x + c₂e^(-2x) + (1/2)x

(for y"+y'-2y=x²)

Given differential equations are:

16y''-8y'+y=0

y"+y'-2y=0

y"+y'-2y = x²

To find the general solution to the given differential equations, we will solve these equations one by one.

(i) 16y'' - 8y' + y = 0

The characteristic equation is:

16m² - 8m + 1 = 0

Solving this quadratic equation, we get m = 1/4, 1/4

Hence, the general solution of the given differential equation is:

y = c₁e^(x/4) + c₂xe^(x/4)..................................................(1)

(ii) y" + y' - 2y = 0

The characteristic equation is:

m² + m - 2 = 0

Solving this quadratic equation, we get m = 1, -2

Hence, the general solution of the given differential equation is:

y = c₁e^x + c₂e^(-2x)..................................................(2)

(iii) y" + y' - 2y = x²

The characteristic equation is:

m² + m - 2 = 0

Solving this quadratic equation, we get m = 1, -2.

The complementary function (CF) of this differential equation is:

y = c₁e^x + c₂e^(-2x)..................................................(3)

Now, we will find the particular integral (PI). Let's assume that the PI of the differential equation is of the form:

y = Ax² + Bx + C

Substituting the value of y in the given differential equation, we get:

2A - 4A + 2Ax² + 4Ax - 2Ax² = x²

Equating the coefficients of x², x, and the constant terms on both sides, we get:

2A - 2A = 1,

4A - 4A = 0, and

2A = 0

Solving these equations, we get

A = 1/2,

B = 0, and

C = 0

Hence, the particular integral of the given differential equation is:

y = (1/2)x²..................................................(4)

The general solution of the given differential equation is the sum of CF and PI.

Hence, the general solution is:

y = c₁e^x + c₂e^(-2x) + (1/2)x²..................................................(5)

Conclusion: Therefore, the general solution of the given differential equations are:

y = c₁e^(x/4) + c₂xe^(x/4) (for 16y''-8y'+y=0)

y = c₁e^x + c₂e^(-2x) (for y"+y'-2y=0)

y = c₁e^x + c₂e^(-2x) + (1/2)x

(for y"+y'-2y=x²)

To know more about differential visit

https://brainly.com/question/13958985

#SPJ11

The particular solution is: y = -1/2 x². The general solution is: y = c1 e^(-2x) + c2 e^(x) - 1/2 x²

The general solution of the given differential equations are:

Given differential equation: 16y'' - 8y' + y = 0

The auxiliary equation is: 16m² - 8m + 1 = 0

On solving the above quadratic equation, we get:

m = 1/4, 1/4

∴ General solution of the given differential equation is:

y = c1 e^(x/4) + c2 x e^(x/4)

Given differential equation: y" + y' - 2y = 0

The auxiliary equation is: m² + m - 2 = 0

On solving the above quadratic equation, we get:

m = -2, 1

∴ General solution of the given differential equation is:

y = c1 e^(-2x) + c2 e^(x)

Given differential equation: y" + y' - 2y = x²

The auxiliary equation is: m² + m - 2 = 0

On solving the above quadratic equation, we get:m = -2, 1

∴ The complementary solution is:y = c1 e^(-2x) + c2 e^(x)

Now we have to find the particular solution, let us assume the particular solution of the given differential equation:

y = ax² + bx + c

We will use the method of undetermined coefficients.

Substituting y in the differential equation:y" + y' - 2y = x²a(2) + 2a + b - 2ax² - 2bx - 2c = x²

Comparing the coefficients of x² on both sides, we get:-2a = 1

∴ a = -1/2

Comparing the coefficients of x on both sides, we get:-2b = 0 ∴ b = 0

Comparing the constant terms on both sides, we get:2c = 0 ∴ c = 0

Thus, the particular solution is: y = -1/2 x²

Now, the general solution is: y = c1 e^(-2x) + c2 e^(x) - 1/2 x²

To know more about differential equations, visit:

https://brainly.com/question/32645495

#SPJ11

Other Questions
the evolution of public health, and what the focus has been over time, can best be described by which flowchart? potential hazard of immune serum globulin, antitoxins, and antivenins would be ___a.) all of these are correntb.) allergic reactionc.) causing the actual disease in an immunocompromised individuald.) mercury poisoning During the early years of the silicon valley, what were some unique qualities of the environments in the start up computer companies? howdoes the peripheral nerve regeneration occur after an injury? Draw stars to represent the relative amounts of proteins on side A and side B of Figure 5.Label Figure 5 with the following terms: "hypertonic", "more solutes", "less water", "hypotonic", "fewer solutes", "more water", semipermeable membrane."Do you think any water molecules move in the opposite direction of the arrow?Upload your sketch below. financial statement frauds and potential of between information providers and users leads to a natural skepticism on the part of users. (enter only one word per blank.) State the beginning reactants and the end products glycolysis, alcoholic fermentation, the citric acid cycle, and the electron transport chain. Describe where these processes take place in the cell and the conditions under which they operate (aerobic or anaerobic), glycolysis: alcoholic fermentation: citric acid cycle: electron transport chain Consider a radioactive sample. Determine the ratio of the number of nuclei decaying during the first half of its halflife to the number of nuclei decaying during the second half of its half-life. An operational amplifier has to be designed for an on-chip audio band pass IGMF filter. Explain using appropriate mathematical derivations what the impact of reducing the input impedance (Zin), and reducing the open loop gain (A) of the opamp will have for the general opamps performance. What effect would any changes to (Zin) or (A) have on the design of an IGMF band pass filter? predict the total packing cost for 25,000 orders, weighing 40,000 pounds, with 4,000 fragile items. round regression intercept to whole dollar and coefficients to two decimal places (nearest cent). enter the final answer rounded to the nearest dollar. Develop a minimum-multiplier realization of a length-7 Type 3 Linear Phase FIR Filter. Pinto LC, Falcetta MR, Rados DV, Leitao CB, Gross JL. Glucagon-like peptide-1 receptor agonists and pancreatic cancer: a meta-analysis with trial sequential analysis. Scientific reports. 2019:9:1-6. originally developed for detecting air pollutants, a technique called proton-induced x-ray emission, which can quickly analyze the chemical elements in almost any substance without destroying it, is finding uses in medicine, archaeology, and criminology. A cylindrical water tank has a fixed surface area of A0.. Find an expression for the maximum volume that such a water tank can take. How does the number 32.4 change when you multiply it by 10 to the power of 2 ? select all that apply. a). the digit 2 increases in value from 2 ones to 2 hundreds. b). each place is multiplied by 1,000 c). the digit 3 shifts 2 places to the left, from the tens place to the thousands place. A(n) ________ account represents the money or investments you have at a brokerage firm. Question 18 options: stock purchasing discretionary brokerage indemnity a _________ is a type of procedure that always returns a value. group of answer choices subprocedure function method event Question 3 Describe the level curves \( L_{1} \) and \( L_{2} \) of the function \( f(x, y)=x^{2}+4 y^{2} \) where \( L_{c}=\left\{(x, y) \in R^{2}: f(x, y)=c\right\} \) Considers F ( x , Y ) = ( 3 + 4 x Y two ) i + 4 x two Y j , C is the arc of the hyperbola Y = one x from the point ( one , one ) until the point ( two , one two ) Determine the potential function using the potential function determine C F d r along curve C. To achieve maximum power transfer between a 44 source and a load ZL (ZL > ZG) using a transmission line with a characteristic impedance of 44 , an inductor with a reactance of 82 is connected in series with the source. Determine the distance from the load, ZL, in terms of wavelengths where the inductor should be connected. Length =