Question

(1 point) For the function f(x)=6x+3sin(x), find all intervals where the function is increasing. f is increasing on (Give your answer as an interval or a list of intervals, e.g., (-infinity, 8] or (1,5),(7,10). Enter none if there are no such intervals.) Similarly, find all intervals where the function is decreasing: f is decreasing on (Give your answer as an interval or a list of intervals, e.g

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(-infinity, 8] or (1,5),(7,10). Enter none if there are no such intervals.) Finally, find all critical points in the graph of f(x) (enter x values as a comma-separated list, or none if there are no critical points):

Answer 1

The function f(x) = 6x + 3sin(x) does not have any intervals where it is increasing or decreasing. There are no critical points in the graph of f(x).

Step-by-step explanation:

To find the intervals where the function f(x) = 6x + 3sin(x) is increasing or decreasing, we need to analyze its derivative. Let's find the derivative first.

The derivative of f(x) = 6x + 3sin(x) is f'(x) = 6 + 3cos(x). To find the intervals where the function is increasing, we need to find where the derivative is positive (greater than zero). Set 6 + 3cos(x) > 0 and solve for x, we have cos(x) > -2. Dividing both sides by 3, we have cos(x) > -2/3. Since the cosine function has a range between -1 and 1, there are no intervals where cos(x) > -2/3, therefore there are no intervals where f(x) is increasing.

To find where the function is decreasing, we need to find where the derivative is negative (less than zero). Set 6 + 3cos(x) < 0 and solve for x, we have cos(x) < -2. Again, since the cosine function has a range between -1 and 1, there are no intervals where cos(x) < -2, therefore there are no intervals where f(x) is decreasing.

The critical points in the graph of f(x) occur when the derivative is equal to zero or undefined. Since the derivative of f(x) = 6x + 3sin(x) is never equal to zero or undefined, there are no critical points in the graph of f(x).

Answer 2

The function f(x) = 6x + 3sin(x) is always increasing for all x in its domain because its derivative f'(x) = 6 + 3cos(x) is always positive. There are no critical points since f'(x) cannot be zero or undefined.

To find where the function f(x) = 6x + 3sin(x) is increasing or decreasing, we will determine its derivative and analyze it. The derivative of f(x), denoted f'(x), represents the rate of change of the function and helps us identify the intervals of growth and decline as well as critical points.

First, let's find the derivative of f(x):

1. The derivative of 6x is 6.
2. The derivative of 3sin(x) is 3cos(x).

So, f'(x) = 6 + 3cos(x).To identify intervals of increase or decrease, we look for values where f'(x) is positive or negative, thus:

• If f'(x)>0, f(x) is increasing.
• If f'(x)<0, f(x) is decreasing.

For f'(x) = 6 + 3cos(x), since the cosine function oscillates between -1 and 1 and the constant is 6, f'(x) will never be negative as the lowest value for 3cos(x) is -3. Therefore, f(x) is always increasing for all x in the domain.

As for critical points, these occur where f'(x) = 0 or is undefined. Since f'(x) cannot be zero due to the constant term and is never undefined, there are no critical points.