Finding K For 137sin²(x) + 4sin(x) + 5 In Interval [k, 5k]
In this comprehensive exploration, we aim to determine the value(s) of k such that the expression 137sin²(x) + 4sin(x) + 5 lies within the interval [k, 5k]. This problem elegantly combines trigonometric functions, quadratic expressions, and interval analysis, demanding a meticulous approach to unravel its intricacies. We will dissect the problem step by step, ensuring clarity and precision in our methodology. By the end of this discussion, you will gain a profound understanding of how to tackle such problems, enhancing your mathematical prowess.
Understanding the Problem
The core of our task lies in analyzing the given trigonometric expression and its range. The expression 137sin²(x) + 4sin(x) + 5 is a quadratic function in terms of sin(x). Since the sine function oscillates between -1 and 1, i.e., -1 ≤ sin(x) ≤ 1, this constraint plays a pivotal role in determining the range of the entire expression. The requirement that the expression's value must lie within the interval [k, 5k] adds another layer of complexity, necessitating us to find suitable values of k that satisfy this condition.
To kickstart our analysis, let's introduce a substitution to simplify the expression. By letting y = sin(x), we transform the expression into a quadratic form, making it easier to handle. This substitution is a common technique in mathematics, allowing us to leverage our understanding of quadratic functions to solve more complex problems. Furthermore, it’s crucial to recognize that the interval [k, 5k] implies that k must be non-negative, i.e., k ≥ 0, because the upper bound 5k cannot be less than the lower bound k. This condition is vital and will guide our solution process.
Transforming the Expression
Let's make the substitution y = sin(x). This substitution transforms the given expression 137sin²(x) + 4sin(x) + 5 into a quadratic function in terms of y. The transformed expression becomes:
137y² + 4y + 5
This quadratic expression is significantly easier to analyze. We need to remember that y, which is sin(x), is constrained to the interval [-1, 1]. This constraint is essential because it limits the possible values that the quadratic expression can take. Without this constraint, the quadratic expression could take any value, which would drastically change the problem. By acknowledging and using this constraint, we can accurately determine the range of the quadratic expression and subsequently find the values of k that satisfy the given condition.
Finding the Range of the Quadratic Expression
To determine the range of the quadratic expression 137y² + 4y + 5 within the interval [-1, 1], we first need to find the vertex of the parabola represented by the quadratic. The vertex gives us the minimum or maximum value of the quadratic function. The y-coordinate of the vertex can be found using the formula:
y_vertex = -b / 2a
where a and b are the coefficients of the quadratic expression ay² + by + c. In our case, a = 137 and b = 4. Plugging these values into the formula, we get:
y_vertex = -4 / (2 * 137) = -2 / 137
Since -2/137 lies within the interval [-1, 1], we need to evaluate the quadratic expression at the vertex and at the endpoints of the interval to determine the minimum and maximum values. Evaluating the quadratic at the vertex y = -2/137 gives us:
137(-2/137)² + 4(-2/137) + 5 = 137(4/137²) - 8/137 + 5 = 4/137 - 8/137 + 5 = 5 - 4/137
This is the minimum value of the quadratic expression within the given interval. Now, we need to evaluate the quadratic at the endpoints of the interval, y = -1 and y = 1. At y = -1:
137(-1)² + 4(-1) + 5 = 137 - 4 + 5 = 138
And at y = 1:
137(1)² + 4(1) + 5 = 137 + 4 + 5 = 146
Therefore, the minimum value of the expression is 5 - 4/137, and the maximum value is 146. The range of the expression within the interval [-1, 1] is approximately [5 - 4/137, 146].
Determining the Values of k
Now that we have the range of the expression 137sin²(x) + 4sin(x) + 5 as approximately [5 - 4/137, 146], we need to find the values of k such that this range lies within the interval [k, 5k]. This means that the minimum value of the expression must be greater than or equal to k, and the maximum value must be less than or equal to 5k. Mathematically, this can be expressed as two inequalities:
- k ≤ 5 - 4/137
- 5k ≥ 146
Let's analyze each inequality separately to find the possible values of k.
Analyzing the First Inequality
The first inequality is:
k ≤ 5 - 4/137
This inequality tells us that k must be less than or equal to 5 - 4/137. We can simplify this expression to get a more precise upper bound for k. Calculating the value, we have:
5 - 4/137 ≈ 4.9708
So, the first inequality gives us the constraint:
k ≤ 4.9708
This inequality provides an upper bound for k. It is essential because it limits how large k can be while still satisfying the condition that the minimum value of the expression is greater than or equal to k. This constraint is crucial in narrowing down the possible values of k and ensuring that our solution is accurate.
Analyzing the Second Inequality
The second inequality is:
5k ≥ 146
To find the lower bound for k, we need to divide both sides of the inequality by 5:
k ≥ 146 / 5
Calculating this value, we get:
k ≥ 29.2
This inequality provides a lower bound for k. It is crucial because it ensures that the maximum value of the expression, which is 146, is less than or equal to 5k. Without this lower bound, the interval [k, 5k] might not be wide enough to contain the range of the expression, leading to an incorrect solution. The constraint k ≥ 29.2 is vital in ensuring that the interval [k, 5k] adequately encompasses the entire range of the expression.
Combining the Inequalities
We have two inequalities:
- k ≤ 4.9708
- k ≥ 29.2
These two inequalities create a contradiction. The first inequality states that k must be less than or equal to approximately 4.9708, while the second inequality states that k must be greater than or equal to 29.2. There is no value of k that can simultaneously satisfy both inequalities. This means that there is no solution to the problem.
The contradiction arises because the range of the expression 137sin²(x) + 4sin(x) + 5, which is approximately [5 - 4/137, 146], cannot be contained within an interval of the form [k, 5k]. The minimum value of the expression is close to 5, while the maximum value is 146. For the interval [k, 5k] to contain this range, k would have to be small enough to be less than the minimum value (approximately 5) and large enough such that 5k is greater than the maximum value (146). These conflicting requirements make it impossible to find a suitable value for k.
Conclusion
After a detailed analysis, we have found that there is no value of k for which the expression 137sin²(x) + 4sin(x) + 5 lies within the interval [k, 5k]. This conclusion is reached by carefully determining the range of the expression and then establishing the constraints on k imposed by the interval [k, 5k]. The conflicting inequalities that arise from these constraints demonstrate the impossibility of finding a suitable k.
This problem serves as an excellent illustration of how mathematical concepts such as trigonometric functions, quadratic expressions, and interval analysis can be combined to create challenging yet insightful problems. The systematic approach we employed, including substitution, finding the range of a quadratic function, and analyzing inequalities, provides a valuable framework for tackling similar problems in the future. Understanding such methodologies not only enhances problem-solving skills but also fosters a deeper appreciation for the interconnectedness of mathematical concepts.
