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barxatty [35]
2 years ago
14

Consider the following functions. f1(x) = x, f2(x) = x2, f3(x) = 2x − 4x2 g(x) = c1f1(x) + c2f2(x) + c3f3(x) Solve for c1, c2, a

nd c3 so that g(x) = 0 on the interval (−[infinity], [infinity]). If a nontrivial solution exists, state it. (If only the trivial solution exists, enter the trivial solution {0, 0, 0}.) {c1, c2, c3} = $$ Correct: Your answer is correct. Determine whether f1, f2, f3 are linearly independent on the interval (−[infinity], [infinity]).
Mathematics
1 answer:
bekas [8.4K]2 years ago
7 0

9514 1404 393

Answer:

  • (c1, c2, c3) = (-2t, 4t, t) . . . . for any value of t
  • NOT linearly independent

Step-by-step explanation:

We want ...

  c1·f1(x) +c2·f2(x) +c3·f3(x) = g(x) ≡ 0

Substituting for the fn function values, we have ...

  c1·x +c2·x² +c3·(2x -4x²) ≡ 0

This resolves to two equations:

  x(c1 +2c3) = 0

  x²(c2 -4c3) = 0

These have an infinite set of solutions:

  c1 = -2c3

  c2 = 4c3

Then for any parameter t, including the "trivial" t=0, ...

  (c1, c2, c3) = (-2t, 4t, t)

__

f1, f2, f3 are NOT linearly independent. (If they were, there would be only one solution making g(x) ≡ 0.)

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A fruit-drinking company tester a new flavor. In a taste test, 34% of the people taking part liked the new flavor. If 51 people
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Answer:

C. 150

Step-by-step explanation:

The number of people who took the test is unknown, so call it x.

34% of x liked the flavor.

51 people liked the flavor.

That means that 34% of x is equal to 51.

34% of x = 51

0.34x = 51

x = 51/0.34

x = 150

Answer: C. 150

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Read 2 more answers
What is equal to n in this equation
IRISSAK [1]

Answer:

n = -45

Step-by-step explanation:

(n-5) /10 = -5

Multiply each side by 10

(n-5) /10 *10= -5*10

n-5 = -50

Add 5 to each side

n-5+5 = -50+5

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4 0
3 years ago
{x^2+6x-27/x^2-9x+18<br> how to solve step by step
Alika [10]

Answer:

Step-by-step explanation:

Step by Step Solution:

More Icon

STEP

1

:

5

Simplify ———————

x2 - 81

Trying to factor as a Difference of Squares:

1.1 Factoring: x2 - 81

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A2 - AB + BA - B2 =

A2 - AB + AB - B2 =

A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 81 is the square of 9

Check : x2 is the square of x1

Factorization is : (x + 9) • (x - 9)

Equation at the end of step

1

:

(((x2)-18x)+81) 5

———————————————•———————————

(((x2)-6x)-27) (x+9)•(x-9)

STEP

2

:

x2 - 18x + 81

Simplify —————————————

x2 - 6x - 27

Trying to factor by splitting the middle term

2.1 Factoring x2 - 18x + 81

The first term is, x2 its coefficient is 1 .

The middle term is, -18x its coefficient is -18 .

The last term, "the constant", is +81

Step-1 : Multiply the coefficient of the first term by the constant 1 • 81 = 81

Step-2 : Find two factors of 81 whose sum equals the coefficient of the middle term, which is -18 .

-81 + -1 = -82

-27 + -3 = -30

-9 + -9 = -18 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -9 and -9

x2 - 9x - 9x - 81

Step-4 : Add up the first 2 terms, pulling out like factors :

x • (x-9)

Add up the last 2 terms, pulling out common factors :

9 • (x-9)

Step-5 : Add up the four terms of step 4 :

(x-9) • (x-9)

Which is the desired factorization

Trying to factor by splitting the middle term

2.2 Factoring x2-6x-27

The first term is, x2 its coefficient is 1 .

The middle term is, -6x its coefficient is -6 .

The last term, "the constant", is -27

Step-1 : Multiply the coefficient of the first term by the constant 1 • -27 = -27

Step-2 : Find two factors of -27 whose sum equals the coefficient of the middle term, which is -6 .

-27 + 1 = -26

-9 + 3 = -6 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -9 and 3

x2 - 9x + 3x - 27

Step-4 : Add up the first 2 terms, pulling out like factors :

x • (x-9)

Add up the last 2 terms, pulling out common factors :

3 • (x-9)

Step-5 : Add up the four terms of step 4 :

(x+3) • (x-9)

Which is the desired factorization

Canceling Out :

2.3 Cancel out (x-9) which appears on both sides of the fraction line.

Equation at the end of step

2

:

(x - 9) 5

——————— • —————————————————

x + 3 (x + 9) • (x - 9)

STEP

3

:

Canceling Out

3.1 Cancel out (x-9) which appears on both sides of the fraction line.

Final result :

5

—————————————————

(x + 3) • (x + 9)

5 0
2 years ago
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