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Black_prince [1.1K]
3 years ago
7

Given the function f(x) = x^4 + 3x^3 - 2x^2 - 6x - 1, use intermediate theorem to decide which of the following intervals contai

ns at least one zero. Select all that apply.

Mathematics
2 answers:
marta [7]3 years ago
8 0

f(x) = x^4 + 3x^3 - 2x^2 - 6x - 1

Lets check with every option

(a) [-4,-3]

We plug in -4  for x  and -3 for x

f(-4) = (-4)^4 + 3(-4)^3 - 2(-4)^2 - 6(-4) - 1= 55

f(-3) = (-3)^4 + 3(-3)^3 - 2(-3)^2 - 6(-3) - 1= -1

f(-4) is positive and f(-3) is negative. there is some value at x=c on the interval [-4,-3] where f(c)=0. so there exists atleast one zero on this interval.

(b) [-3,-2]

We plug in -3  for x  and -2 for x

f(-3) = (-3)^4 + 3(-3)^3 - 2(-3)^2 - 6(-3) - 1= -1

f(-2) = (-2)^4 + 3(-2)^3 - 2(-2)^2 - 6(-2) - 1= -5

f(-2) is negative and f(-3) is negative. there is no value at x=c on the interval [-3,-2] where f(c)=0.  

(c) [-2,-1]

We plug in -2  for x  and -1 for x

f(-2) = (-2)^4 + 3(-2)^3 - 2(-2)^2 - 6(-2) - 1= -5

f(-1) = (-1)^4 + 3(-1)^3 - 2(-1)^2 - 6(-1) - 1= 1

f(-2) is negative and f(-1) is positive. there is some value at x=c on the interval [-2,-1] where f(c)=0. so there exists atleast one zero on this interval.

(d) [-1,0]

We plug in -1  for x  and 0 for x

f(-1) = (-1)^4 + 3(-1)^3 - 2(-1)^2 - 6(-1) - 1= 1

f(0) = (0)^4 + 3(0)^3 - 2(0)^2 - 6(0) - 1= -1

f(-1) is positive and f(0) is negative. there is some value at x=c on the interval [-1,0] where f(c)=0. so there exists atleast one zero on this interval.

(e) [0,1]

We plug in 0  for x  and 1 for x

f(0) = (0)^4 + 3(0)^3 - 2(0)^2 - 6(0) - 1= -1

f(1) = (1)^4 + 3(1)^3 - 2(1)^2 - 6(1) - 1= -5

f(0) is negative and f(1) is negative. there is no value at x=c on the interval [0,1] where f(c)=0.  

(f) [1,2]

We plug in 1  for x  and 2 for x

f(1) = (1)^4 + 3(1)^3 - 2(1)^2 - 6(1) - 1= -5

f(2) = (2)^4 + 3(2)^3 - 2(2)^2 - 6(2) - 1= 19

f(-4) is positive and f(-3) is negative. there is some value at x=c on the interval [-4,-3] where f(c)=0. so there exists atleast one zero on this interval.

so answers are (a) [-4,-3], (c) [-2,-1],  (d) [-1,0], (f) [1,2]

Wittaler [7]3 years ago
4 0

Answer:

(a). [-4, -3]

(c). [-2, -1]

(d). [-1, 0]

(f). [1, 2]

Step-by-step explanation:

The Intermediate Value Theorem states that, for a function f(x) which is continuous in [a, b] also f(a) and f(b) has opposite signs, then there must be a point c lies in interval (a, b) such that f(c) = 0.

we have function: f(x) = x⁴ + 3x³ - 2x² - 6x - 1

(a) [-4, -3]: Calculating

f(-4) = (-4)⁴ + 3(-4)³ - 2(-4)² - 6×-4 - 1 = 55

f(-3) = -1

both has opposite sign so there must be at least one zero lies in the interval [-4, -3]

(b) [-3, -2]

f(-3) = -1

f(-2) = -5

Since both has same sign, Hence no zeros lies in this interval.

(c) [-2, -1]

f(-2) = -5

and f(-1) = 1

Since both has opposite sign hence at least one zeroes lies in this interval.

(d) [-1, 0]

f(-1) = 1

f(0) = -1

Since both has opposite sign hence at least one zeroes lies in this interval.

(e) [0, 1]

f(0) = -1

f(1) = -5

Since both has same sign, Hence no zeros lies in this interval.

(f) [1, 2]

f(1) = -5

f(2) = 19

Since both has opposite sign hence at least one zeroes lies in this interval.

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Answer: 3, 4,5 and 17, 15,8

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Step-by-step explanation:

In order for the measures of the legs and hypotenuse given to form a right angle triangle, they must be Pythagorean triples. A Pythagoras triple is a set of numbers that perfectly satisfy the Pythagorean theorem. The Pythagorean theorem is expressed as

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1) 3, 4, 5

5² = 3² + 4² = 9 + 16

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2) 4, 11, 14

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196 = 137

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3) 9, 14, 17

17² = 14² + 9² = 196 + 81

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4) 8, 14, 16

16² = 14² + 8² = 196 + 64

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8 0
3 years ago
Evaluate i^31 please help
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ANSWER

{i}^{31}  =  - i

EXPLANATION

We want to evaluate

{i}^{31}

Use indices to rewrite the expression as:

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We know that

{i}^{2}  =  - 1

So we rewrite the expression to obtain;

=  ({ {i}^{2}) }^{15}  \times i

This gives us;

=   {( - 1) }^{15}  \times i

This simplifies to

=  - 1 \times i

=  - i

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2 years ago
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The trigonometric ratios show that the angle FHE is 48.59°.

<h3>RIGHT TRIANGLE</h3>

A triangle is classified as a right triangle when it presents one of your angles equal to 90º.  The greatest side of a right triangle is called hypotenuse. And, the other two sides are called cathetus or legs.

The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem.

The Pythagorean Theorem says: (hypothenuse)^2=(leg_1)^2+(leg_2)^2. And the main trigonometric ratios are:

sin(\alpha) =\frac{opposite \;leg }{hypotenuse} \\ \\ cos(\alpha) =\frac{adjacent\;leg }{hypotenuse} \\ \\ tan(\alpha) =\frac{sin(\alpha )}{cos(\alpha )}= \frac{opposite \;leg }{adjacent\;leg } \\ \\

It is important to remember that the sum of internal angles for any triangle is 180°.

From the question, it is possible to see 2 right triangles (HGF and FHE).

You can find the hypotenuse of the triangle HGF from the  trigonometric ratio: sen Θ

sin45=\frac{opposite\; leg }{hypotenuse} =\frac{\sqrt8}{hypotenuse}\\ \\ \frac{\sqrt{2} }{2} =\frac{\sqrt{8} }{hypotenuse} \\ \\ \sqrt{2}*hypotenuse=2\sqrt{8} \\ \\ hypotenuse=\frac{2\sqrt{8} }{\sqrt{2}} =2\sqrt{4} =2*2=4

The hypotenuse of triangle HGF is one of legs for the triangle FHE. The, you can find the angle FHE from the  trigonometric ratio: tan β. Thus,

sin \beta =\frac{opposite\; leg }{adjacent\; leg} =\frac{3}{4}\\ \\ sin \beta=\frac{3}{4}=0.84806\\ \\ arcsin\beta =48.59^{\circ \:}

Learn more about trigonometric ratios here:

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