Which of the following polynomials is guaranteed by the Intermediate Value Theorem to have a zero on the interval [3,4]?

j(x) = 2x² - 5x - 5

General Formulas and Concepts:

Calculus

Step-by-step explanation:

Step 1: Define functions

f(x) = 2x² + x - 3

g(x) = -4x² + 3x - 4

h(x) = x³ - 2x² - x + 4

j(x) = 2x² - 5x - 5

Step 2: Check

Check if each function is continuous over the interval [3, 4].

1. f(x) = 2x² + x - 3

  (a) f is cont on [3, 4]? Yes

2. g(x) = -4x² + 3x - 4

  (a) f is cont on [3, 4]? Yes

3. h(x) = x³ - 2x² - x + 4

  (a) f is cont on [3, 4]? Yes

4. j(x) = 2x² - 5x - 5

  (a) f is cont on [3, 4]? Yes

Step 3: IVT

Find the zeros on the functions in the interval [3, 4].

1. f(x) = 2x² + x - 3

  (b) f(3) = 18 < k = 0 < f(4) = 33? No

2. g(x) = -4x² + 3x - 4

  (b) f(3) = -31 < k = 0 < f(4) = -56? No

3. h(x) = x³ - 2x² - x + 4

  (b) f(3) = 10 < k = 0 < f(4) = 32? No

4. j(x) = 2x² - 5x - 5

  (b) f(3) = -2 < k = 0 < f(4) = 7? Yes

Step 4: Justify IVT Function

j(x) = 2x² - 5x - 5

  (a) f is cont on [3, 4]? Yes

  (b) f(3) = -2 < k = 0 < f(4) = 7? Yes

  ∴ by IVT, there exists a c∈[3, 4] such that f(c) = 0.

5x+40

5(x + 8)

7x - 49

7(x - 7)

18x + 9

9(2x + 1)

15x - 10

5(3x - 2)

step-by-step explanation:

for number one all you have to do is factor 5 out of 5x + 40

for number 2 factor 7 out of 7x - 49

for number three factor 9 out of 18x + 9

for number four factor 5 out of 15x - 10

10 you would take 50/ .2

step-by-step explanation: