Solved on Dec 15, 2023

Determine the minimum and maximum value of $f(x) = -8 \cos 2x$ .

STEP 1

Assumptions
1. The function given is $f(x)=-8 \cos 2x$ .
2. The domain of $f(x)$ is all real numbers, as it is a trigonometric function.
3. We know that the range of the cosine function $\cos x$ is between -1 and 1, inclusive.

STEP 2

We need to understand the behavior of the cosine function to find the minimum and maximum values of $f(x)$ . The cosine function oscillates between -1 and 1, and it is periodic with a period of $2\pi$ .

STEP 3

Since $f(x)$ is a transformation of the cosine function, we need to determine how the transformation affects the range of the original cosine function.

STEP 4

The transformation involves multiplying the cosine function by -8. This will affect the amplitude of the cosine function.

STEP 5

The amplitude of the cosine function is the maximum distance from the horizontal axis (which is 1 for $\cos x$ ). Multiplying by -8 will change the amplitude to 8.

STEP 6

Since the cosine function oscillates between -1 and 1, multiplying by -8 will change this range to oscillate between $8$ and $-8$ .

STEP 7

The negative sign in front of the 8 will also reflect the graph of the cosine function across the horizontal axis. However, this reflection does not change the absolute values of the maximum and minimum; it only changes their signs.

STEP 8

Now we can determine the maximum and minimum values of $f(x)$ . The maximum value of $f(x)$ will be when $\cos 2x$ is at its minimum, which is -1.

STEP 9

Calculate the maximum value of $f(x)$ by substituting -1 into the function for $\cos 2x$ .
$f(x) = -8 \cdot (-1)$

STEP 10

Calculate the result to find the maximum value.
$f(x)_{max} = 8$

STEP 11

The minimum value of $f(x)$ will be when $\cos 2x$ is at its maximum, which is 1.

STEP 12

Calculate the minimum value of $f(x)$ by substituting 1 into the function for $\cos 2x$ .
$f(x) = -8 \cdot 1$

STEP 13

Calculate the result to find the minimum value.
$f(x)_{min} = -8$
The minimum value of the function $f(x)=-8 \cos 2x$ is -8, and the maximum value is 8.

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Determine the minimum and maximum value of $f(x) = -8 \cos 2x$ .

STEP 1

Assumptions
1. The function given is $f(x)=-8 \cos 2x$ .
2. The domain of $f(x)$ is all real numbers, as it is a trigonometric function.
3. We know that the range of the cosine function $\cos x$ is between -1 and 1, inclusive.

STEP 2

We need to understand the behavior of the cosine function to find the minimum and maximum values of $f(x)$ . The cosine function oscillates between -1 and 1, and it is periodic with a period of $2\pi$ .

STEP 3

Since $f(x)$ is a transformation of the cosine function, we need to determine how the transformation affects the range of the original cosine function.

STEP 4

The transformation involves multiplying the cosine function by -8. This will affect the amplitude of the cosine function.

STEP 5

The amplitude of the cosine function is the maximum distance from the horizontal axis (which is 1 for $\cos x$ ). Multiplying by -8 will change the amplitude to 8.

STEP 6

Since the cosine function oscillates between -1 and 1, multiplying by -8 will change this range to oscillate between $8$ and $-8$ .

STEP 7

The negative sign in front of the 8 will also reflect the graph of the cosine function across the horizontal axis. However, this reflection does not change the absolute values of the maximum and minimum; it only changes their signs.

STEP 8

Now we can determine the maximum and minimum values of $f(x)$ . The maximum value of $f(x)$ will be when $\cos 2x$ is at its minimum, which is -1.

STEP 9

Calculate the maximum value of $f(x)$ by substituting -1 into the function for $\cos 2x$ .
$f(x) = -8 \cdot (-1)$

STEP 10

Calculate the result to find the maximum value.
$f(x)_{max} = 8$

STEP 11

The minimum value of $f(x)$ will be when $\cos 2x$ is at its maximum, which is 1.

STEP 12

Calculate the minimum value of $f(x)$ by substituting 1 into the function for $\cos 2x$ .
$f(x) = -8 \cdot 1$

STEP 13

Calculate the result to find the minimum value.
$f(x)_{min} = -8$
The minimum value of the function $f(x)=-8 \cos 2x$ is -8, and the maximum value is 8.

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Determine the minimum and maximum value of f(x)=−8cos⁡2xf(x) = -8 \cos 2xf(x)=−8cos2x.

STEP 1

Assumptions1. The function given is f(x)=−8cos⁡2x f(x)=-8 \cos 2x f(x)=−8cos2x.2. The domain of f(x) f(x) f(x) is all real numbers, as it is a trigonometric function.3. We know that the range of the cosine function cos⁡x \cos x cosx is between -1 and 1, inclusive.

STEP 2

We need to understand the behavior of the cosine function to find the minimum and maximum values of f(x) f(x) f(x). The cosine function oscillates between -1 and 1, and it is periodic with a period of 2π 2\pi 2π.

STEP 3

Since f(x) f(x) f(x) is a transformation of the cosine function, we need to determine how the transformation affects the range of the original cosine function.

STEP 4

The transformation involves multiplying the cosine function by -8. This will affect the amplitude of the cosine function.

STEP 5

The amplitude of the cosine function is the maximum distance from the horizontal axis (which is 1 for cos⁡x \cos x cosx). Multiplying by -8 will change the amplitude to 8.

STEP 6

Since the cosine function oscillates between -1 and 1, multiplying by -8 will change this range to oscillate between 8 8 8 and −8 -8 −8.

STEP 7

The negative sign in front of the 8 will also reflect the graph of the cosine function across the horizontal axis. However, this reflection does not change the absolute values of the maximum and minimum; it only changes their signs.

STEP 8

Now we can determine the maximum and minimum values of f(x) f(x) f(x). The maximum value of f(x) f(x) f(x) will be when cos⁡2x \cos 2x cos2x is at its minimum, which is -1.

STEP 9

Calculate the maximum value of f(x) f(x) f(x) by substituting -1 into the function for cos⁡2x \cos 2x cos2x.f(x)=−8⋅(−1) f(x) = -8 \cdot (-1) f(x)=−8⋅(−1)

STEP 10

Calculate the result to find the maximum value.f(x)max=8 f(x)_{max} = 8 f(x)max​=8

STEP 11

The minimum value of f(x) f(x) f(x) will be when cos⁡2x \cos 2x cos2x is at its maximum, which is 1.

STEP 12

Calculate the minimum value of f(x) f(x) f(x) by substituting 1 into the function for cos⁡2x \cos 2x cos2x.f(x)=−8⋅1 f(x) = -8 \cdot 1 f(x)=−8⋅1

STEP 13

Calculate the result to find the minimum value.f(x)min=−8 f(x)_{min} = -8 f(x)min​=−8The minimum value of the function f(x)=−8cos⁡2x f(x)=-8 \cos 2x f(x)=−8cos2x is -8, and the maximum value is 8.

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Determine the minimum and maximum value of $f(x) = -8 \cos 2x$ .

Assumptions
1. The function given is $f(x)=-8 \cos 2x$ .
2. The domain of $f(x)$ is all real numbers, as it is a trigonometric function.
3. We know that the range of the cosine function $\cos x$ is between -1 and 1, inclusive.

We need to understand the behavior of the cosine function to find the minimum and maximum values of $f(x)$ . The cosine function oscillates between -1 and 1, and it is periodic with a period of $2\pi$ .

Since $f(x)$ is a transformation of the cosine function, we need to determine how the transformation affects the range of the original cosine function.

The amplitude of the cosine function is the maximum distance from the horizontal axis (which is 1 for $\cos x$ ). Multiplying by -8 will change the amplitude to 8.

Since the cosine function oscillates between -1 and 1, multiplying by -8 will change this range to oscillate between $8$ and $-8$ .

Now we can determine the maximum and minimum values of $f(x)$ . The maximum value of $f(x)$ will be when $\cos 2x$ is at its minimum, which is -1.

Calculate the maximum value of $f(x)$ by substituting -1 into the function for $\cos 2x$ .
$f(x) = -8 \cdot (-1)$

Calculate the result to find the maximum value.
$f(x)_{max} = 8$

The minimum value of $f(x)$ will be when $\cos 2x$ is at its maximum, which is 1.

Calculate the minimum value of $f(x)$ by substituting 1 into the function for $\cos 2x$ .
$f(x) = -8 \cdot 1$

Calculate the result to find the minimum value.
$f(x)_{min} = -8$
The minimum value of the function $f(x)=-8 \cos 2x$ is -8, and the maximum value is 8.