Write the Maximum and Minimum values of cos (cos x).
Write the Maximum and Minimum values of cos (cos x).
Aniket Singh , 5 Months ago
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11 grade maths others> Write the Maximum and Minimum values of c...
1 AnswersAskiitians Tutor Team
Last Activity: 5 Months ago
To find the maximum and minimum values of cos(cos(x)), let's proceed step by step.
Step 1: Understand the function
We are given the function f(x) = cos(cos(x)). This is a composition of two functions:
The outer function is cos(u), where u = cos(x).
The inner function is cos(x).
Step 2: Analyze the range of cos(x)
The range of cos(x) is between -1 and 1. That means: -1 ≤ cos(x) ≤ 1.
Step 3: Analyze the behavior of cos(u)
Now, we need to understand how the function cos(u) behaves when u is within the range [-1, 1].
The function cos(u) is decreasing in the interval from 0 to π.
At u = -1, cos(-1) ≈ 0.5403.
At u = 1, cos(1) ≈ 0.5403.
The maximum value of cos(u) occurs when u = 0, which is cos(0) = 1.
Step 4: Determine the maximum and minimum values of cos(cos(x))
Since cos(x) takes values between -1 and 1, we can evaluate cos(cos(x)) at the endpoints of this interval:
When cos(x) = -1, cos(cos(x)) = cos(-1) ≈ 0.5403.
When cos(x) = 1, cos(cos(x)) = cos(1) ≈ 0.5403.
When cos(x) = 0, cos(cos(x)) = cos(0) = 1.
Therefore, the maximum value of cos(cos(x)) is 1, and the minimum value is approximately 0.5403.
Final Answer:
Maximum value of cos(cos(x)) = 1.
Minimum value of cos(cos(x)) ≈ 0.5403.
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