Find the minimum or maximum value of the function. Describe the domain and range of the function, and where the function is increasing or decreasing. h(x)=−3x2−6x+1 Answer A The maximum value is 4. The domain is all real numbers and the range is y≤4 . The function is increasing to the left of x=−1 and decreasing to the right of x=−1 . B The minimum value is 4. The domain is all real numbers and the range is y≥4 . The function is increasing to the left of x=−1 and decreasing to the right of x=−1 . C The maximum value is −8 . The domain is all real number and the range is y≤−8 . The function is increasing to the left of x=−1 and decreasing to the right of x=−1 . D The minimum value is −8 . The domain is all real numbers and the range is y≥−8 . The function is increasing to the left of x=−1 and decreasing to the right of x=−1 .

The maximum value is 4.
The domain is all real numbers and the range is y≤4 .
The function is increasing to the left of x=−1 and decreasing to the right of x=−1 .

From the question, we have the following parameters that can be used in our computation:

h(x) =-3x² - 6x + 1

Differentiate

h'(x) = -6x - 6

Set to 0

-6x - 6 = 0

So, we have

-6x = 6

Divide by -6

x = -1

Substitute x = -1 in h(x) =-3x² - 6x + 1

h(-1) =-3(-1)² - 6(-1) + 1

Evaluate

h(-1) = 4

The leading coefficient is negative

So, the vertex is maximum

This means that the maximum value is 4

The function is a quadratic function

So, the domain is all real numbers

In (a), we have

The leading coefficient is negative

So, the vertex is maximum

This means that the range is y ≤ 4

In (a), we have

x = -1

The vertex is maximum

This means that

The function increases at the left of x = −1 and decreases at the right of x = −1 .

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