Which of the following best corresponds to the x-value where the function f(x)=cos x is a minimum in the interval x = [25,30]?
A) 25.1
B)26.7
C)28.3
D)29.8
Can someone show me how to solve this step by step? Thanks :)
you know that cos(x) = -1 at x = π,3π,5π, ...
So, we can check
7π = 21.99
9π = 28.27
Looks like C to me
To find the x-value where the function f(x) = cos(x) is a minimum in the interval x = [25, 30], we need to analyze the behavior of the function within that interval.
Step 1: Determine the critical points of the function. Critical points are the points where the derivative of the function is either zero or undefined. In this case, we have f(x) = cos(x), and its derivative is f'(x) = -sin(x). Set f'(x) = 0 and solve for x:
-sin(x) = 0
Since sin(x) = 0 when x = 0, π, 2π, etc., we need to find the values of x within the interval [25, 30] that satisfy this condition.
Step 2: Determine the values of x that satisfy the condition within the interval. Since the interval is [25, 30], we can narrow down the possible values to x = 25, and x = 30. Plug these values into the derivative to see if they yield a minimum:
f'(25) = -sin(25) ≈ -0.132
f'(30) = -sin(30) ≈ -0.499
Both values are negative, indicating a decreasing slope. Since the question asks for a minimum, we need to look for a value where the slope is changing from negative to positive.
Step 3: Analyze the behavior of the function. The function f(x) = cos(x) is a continuous function with periodic behavior. The cosine function has a period of 2π, meaning it repeats every 2π units. This means that f(x) will have a minimum point whenever its value reaches the lowest value within the interval [25, 30].
Step 4: Find the x-value that corresponds to the lowest value of f(x). To do this, we can evaluate the function at the endpoints of the interval, as well as the critical points found in step 2:
f(25) = cos(25) ≈ 0.991
f(30) = cos(30) ≈ 0.154
From these values, we can see that f(25) is the smallest value within the interval [25, 30].
Therefore, the x-value where the function f(x) = cos(x) is a minimum in the interval x = [25, 30] is x = 25.
Answer: A) 25.1
To find the minimum value of the function f(x) = cos(x) in the given interval [25,30], we need to find the x-value at which the function reaches its minimum. Here's how you can solve this step by step:
1. Start by evaluating the function at the interval's endpoints:
- Calculate f(25) = cos(25) and f(30) = cos(30).
2. Use a calculator or a trigonometric table to find the numeric values of cos(25) and cos(30).
3. Compare these values to determine which one is smaller.
- If f(25) < f(30), then the x-value 25 corresponds to a minimum.
- If f(30) < f(25), then the x-value 30 corresponds to a minimum.
4. Finally, choose the option that matches the x-value you found in step 3.
- If 25 corresponds to a minimum, choose option A) 25.1.
- If 30 corresponds to a minimum, choose option D) 29.8.
Remember that the cosine function oscillates between -1 and 1, so the minimum value of cos(x) within the given interval will be the smallest value it reaches.