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LenaWriter [7]
3 years ago
6

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

Mathematics
2 answers:
nadezda [96]3 years ago
5 0

Answer:

c =  1

Step-by-step explanation:

Given:-

- The given function f(x) is:

                                f(x) = 3x^2 - 2x + 1 ,   [ 0 , 2 ]

Find:-

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

f it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem.

Solution:

- The mean value theorem states that if a function f(x) is differentiable over the range [ x1 , x2 ] , then there exist a value c within the range [ x1 , x2 ]. Such that:

                                f'(c) = [ f(x2) - f(x1) ] / [ x2 - x1 ]

- Note: The right hand side of above theorem expresses the " Secant " line.

- We see that the function f(x) is a polynomial function with degree 2 which is continuous and differentiable over the entire interval of real numbers R. For which the differential of the given function is:

                                f'(x) = 6x - 2

- It exist for all real value of x and is continuous ( Linear Line ).

- It satisfies the hypothesis of the mean value theorem. So our function f(x) to be differentiable over the range [ 0 , 2 ]. then there exist a value c within the range [ 0 , 2 ]

                                f'(x) = [ f(2) - f(0) ] / [ 2 - 0 ]

                                f'(x) = [ 3(2)^2 - 2(2) + 1 - 1 ] / [ 2 - 0 ]

                                f'(x) = [ 8 ] / [ 2 ] = 4

                                f'(c) = 6c - 2 = 4

                                c = 6 / 6 = 1

- Hence, the required value of c = 1.

liq [111]3 years ago
4 0

Answer:

7/3

Step-by-step explanation:

f(x) = 3x^2 + 2x + 1, [0, 2]

f(0)=1

f(2)=17

f'(c)=f(2)-f(0)/2-1

f'(c)=17-1/1=16

f'(c)= 6x+2

6x+2=16

6x=16-2

6x=14

x=14/6

x=7/3

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aleksklad [387]

56 ÷ 8 = 7

8 goes into 56 7 times

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3 years ago
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what is the rate of change for the linear relationship modeled in the table? x y −1 10 1 9 3 8 5 7 -1/2 0 1/2 2
Vesnalui [34]

Answer:

Rate of change for the linear relationship modeled is \dfrac{-1}{2}

Step-by-step explanation:

As the there is a linear relationship in the points, so all these points will be on a single straight line. Hence the slope will be same throughout all the points.

We know that, the slope of the line joining (x₁, y₁) and (x₂, y₂) is,

m=\dfrac{y_2-y_1}{x_2-x_1}

Putting the points as (-1, 10) and (1, 9), we get

m=\dfrac{9-10}{1-(-1)}

=\dfrac{9-10}{1+1}

=\dfrac{-1}{2}

Rate of change is the slope of the line joining all these points.

7 0
3 years ago
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Two sides of an acute triangle meaure 5 inches and 8 inches. The length of the longest side is unknown. What is the greatest pos
gladu [14]

The greatest whole possible whole number length of the unknown side is 9 inches.

<h3>How to identify if a triangle is acute?</h3>

Let us have:

H = biggest side of the triangle

And let we get  A and B as rest of the two sides.

Then we get:

If

A^2 + B^2 < C^2

then the triangle is acute

Two sides of an acute triangle measure as 5 inches and 8 inches

The length of the longest side is unknown.

We have to find the length of the unknown side

WE know that the longest side of any triangle is a hypotenuse

For an acute triangle we know:

A^2 + B^2 < C^2

Here in this sum,

a = 5 inches

b = 8 inches

c = ?

Substituting we get,

A^2 + B^2 < C^2\\\\5^2 + 8^2 < C^2\\\\25 + 64= C^2

c < 9

Hence, The greatest whole possible whole number length of the unknown side is 9 inches.

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3 0
2 years ago
Please help me solve this I am stuck I will mark brainliest
fomenos

Answer:

the answer is no its not the solution

Step-by-step explanation:

6 0
2 years ago
Find the solutions of the quadratic equation 14x^2+9x+10=014x
Vesnalui [34]

Answer:

Option B. x=-\frac{9}{28}(+/-)\frac{\sqrt{479}}{28}i

Step-by-step explanation:

we know that

The formula to solve a quadratic equation of the form ax^{2} +bx+c=0 is equal to

x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}

in this problem we have

14x^{2}+9x+10=0

so

a=14\\b=9\\c=10

substitute in the formula

x=\frac{-9(+/-)\sqrt{9^{2}-4(14)(10)}} {2(14)}

x=\frac{-9(+/-)\sqrt{-479}} {28}

Remember that

i=\sqrt{-1}

substitute

x=\frac{-9(+/-)\sqrt{479}i} {28}  

x=-\frac{9}{28}(+/-)\frac{\sqrt{479}}{28}i

5 0
3 years ago
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