(v) F(x)=(sqrt(cos X-(1)/(2)))/(sqrt(6+35x-6x^(2)))

5 min read Jun 16, 2024

(v) F(x)=(sqrt(cos X-(1)/(2)))/(sqrt(6+35x-6x^(2)))

Analyzing the Function f(x) = (√(cos x - 1/2)) / (√(6 + 35x - 6x²))

This article aims to analyze the function f(x) = (√(cos x - 1/2)) / (√(6 + 35x - 6x²)). We will explore its domain, range, critical points, and other significant features.

Domain

The domain of a function is the set of all possible input values (x) for which the function is defined. To determine the domain of our function, we need to consider the following restrictions:

Radicands (expressions under the square roots) must be non-negative.
- √(cos x - 1/2): This requires cos x - 1/2 ≥ 0. Solving this inequality, we get cos x ≥ 1/2. This holds true for x within the intervals [-π/3 + 2πk, π/3 + 2πk], where k is an integer.
- √(6 + 35x - 6x²): This requires 6 + 35x - 6x² ≥ 0. Factoring the quadratic expression, we get (2x - 1)(3x - 6) ≤ 0. Solving this inequality, we find the solution is x ∈ [1/2, 2].
Denominator cannot be zero.
- √(6 + 35x - 6x²) = 0 implies 6 + 35x - 6x² = 0. Solving this quadratic equation, we get x = 1/2 or x = 2. However, these values are already excluded due to the previous restriction.

Combining these restrictions, the domain of the function f(x) is: [-π/3 + 2πk, π/3 + 2πk] ∩ [1/2, 2], where k is an integer. This represents intervals within which both radicands are non-negative.

Range

Determining the exact range of this function is a complex task. However, we can identify some key properties:

Boundedness: Due to the cosine function, the numerator √(cos x - 1/2) has a maximum value of √(1/2) and a minimum value of 0. The denominator √(6 + 35x - 6x²) is always positive within the domain. Therefore, the function itself is also bounded.
Asymptotes: The function does not have any vertical asymptotes as the denominator is never zero within the domain. Additionally, there are no horizontal asymptotes as the function oscillates between positive and negative values.

Critical Points

Critical points are points where the derivative of the function is zero or undefined. To find them, we need to differentiate f(x) and analyze where the derivative is zero or undefined. This involves applying the chain rule and quotient rule for differentiation.

Derivative: After applying these rules, the derivative of f(x), denoted as f'(x), will be a complex expression involving trigonometric and polynomial terms.
Zeros: Setting f'(x) = 0 and solving for x will give us potential critical points. However, the solution may not be analytically obtainable, requiring numerical methods.
Undefined points: The derivative will be undefined where the denominator is zero. This will coincide with the boundaries of the domain.

Conclusion

The function f(x) = (√(cos x - 1/2)) / (√(6 + 35x - 6x²)) exhibits a complex behavior due to the combination of trigonometric and polynomial terms. Its domain is restricted by the need for non-negative radicands, resulting in intervals within the domain. The range is bounded, but its exact boundaries are challenging to determine analytically. Finding critical points requires analyzing the derivative, which can be a complex task. Further investigation through numerical methods and graphical analysis can provide more insight into the behavior of this function.