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sp2606 [1]
3 years ago
12

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that s

atisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.) f(x) = cos(5x), [π/20, 7π/20] c =
Mathematics
1 answer:
myrzilka [38]3 years ago
8 0

Answer:

There are 2 points where f'(x) = 0

At x = 0, and at x = 5/π

Step-by-step explanation:

Rolle's theorem states that for any differentiable function on the interval {a, b}, if f(a) = f(b), then there is at least one point in the interval where f'(c) = 0

Evaluate the end points of the interval to see if we can apply Rolle's Theorem...

f(π/20) = cos (5π/20) = cos (π/4) = (√2)/2

f(7π/20) = cos [5(7π]/20) = cos (35π/20) = cos (7π/4) = (√2)/2

So by Rolle's Theorem, there will be at least one point where f'(c) = 0, so find f'(x)

f'(x) = -5sin(5x)

find where this equal zero...

0 = -5sin(5x)

0 = sin(5x)

Sin x = 0 at x = 0, and x = π, so we have..

5x = 0,  so x = 0

5x = π, so x = 5/π

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Please help me ASAP 50 points
Arte-miy333 [17]

Answer:

x=2

Vertex (2,-8)

Intercepts: 2 ±2/3 sqrt(6)

Because the parabola opens upwards, it has a minimum at the vertex

Domain:  all real numbers

Range: y>= -8

Step-by-step explanation:

y= 3x^2 -12x+4

To find the axis of symmetry, we use the formula

h= -b/2a  where ax^2 +bx+c

h = -(-12)/2*3 = 12/6=2

The axis of symmetry is x=2

The x coordinate of the vertex is at the axis of symmetry. To find the y coordinate, substitute x into the equation

y= 3(2)^2 -12(2) +4

y = 3*4-24+4

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  = -8

The vertex is at (2,-8)

We can use the quadratic formula to find the x intercepts

x = -b ±sqrt(b^2-4ac)

    -----------------------

           2a

  =  12 ± sqrt(12^2 - 4*3*4)

  ---------------------------------

    2*3

=   12± sqrt(144-48)

 ----------------------

  6

= 12±sqrt(96)

---------------

6

=2 ±1/6 * 4sqrt(6)

= 2 ±2/3 sqrt(6)


Because the parabola opens upwards, it has a minimum at the vertex.

The domain is the values that x can take.

X can be all real numbers

The range is the values that y can take.  Since there is a minimum, y must be greater than that minimum

Range: y>= -8

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