The Minimum Value of the Function

Oct 13, 2025 by windailys.com 85 views

Introduction

In mathematics, functions are used to describe the relationship between variables. The function y = a * cos(bx) is a type of trigonometric function that is used to model various real-world phenomena, such as the motion of a pendulum or the vibration of a spring. The minimum value of a function is the lowest point on its graph, and it is an important concept in calculus and optimization. In this article, we will discuss the minimum value of the function y = a * cos(bx) and determine the value of x at which it occurs.

**The Function y = a * cos(bx)**

The function y = a * cos(bx) is a cosine function with amplitude a and frequency b. The amplitude a is a measure of the maximum value of the function, while the frequency b determines the number of cycles of the function in a given interval. The function can be written in the form:

y = a * cos(bx)

where a is a constant, b is a positive constant, and x is the independent variable.

The Minimum Value of the Function

The minimum value of the function y = a * cos(bx) occurs when the cosine function is at its minimum value, which is -1. This means that the function y = a * cos(bx) will have a minimum value when:

cos(bx) = -1

Solving for x

To find the value of x at which the minimum value of the function occurs, we need to solve the equation:

cos(bx) = -1

Using the properties of the cosine function, we know that the cosine function has a minimum value of -1 at odd multiples of π/2. Therefore, we can write:

bx = (2n + 1) * π/2

where n is an integer.

Solving for x in Terms of b

To find the value of x in terms of b, we can divide both sides of the equation by b:

x = (2n + 1) * π/2b

Given Value of a

We are given that:

a = 1300* (10 + 35/b)

Substituting the Value of a

Substituting the value of a into the equation for the minimum value of the function, we get:

y = 1300* (10 + 35/b) * cos(bx)

Finding the Minimum Value of the Function

To find the minimum value of the function, we need to find the value of x that minimizes the function. This occurs when:

cos(bx) = -1

Solving for x

Using the equation:

bx = (2n + 1) * π/2

we can solve for x:

x = (2n + 1) * π/2b

Finding the Value of n

To find the value of n, we need to find the smallest integer value of n that satisfies the equation:

(2n + 1) * π/2b < 2π

Solving for n

Solving for n, we get:

n < (4b - 1) * π/2π

Simplifying the Inequality

Simplifying the inequality, we get:

n < 2b - 1/2

Finding the Smallest Integer Value of n

The smallest integer value of n that satisfies the inequality is:

n = 2b - 1

Finding the Value of x

Substituting the value of n into the equation for x, we get:

x = (2(2b - 1) + 1) * π/2b

Simplifying the Equation

Simplifying the equation, we get:

x = (4b - 1) * π/2b

Simplifying the Equation Further

Simplifying the equation further, we get:

x = 2π/2 - π/2b

Simplifying the Equation Even Further

Simplifying the equation even further, we get:

x = π/2 - π/2b

Simplifying the Equation One Last Time

Simplifying the equation one last time, we get:

x = π/2(1 - 1/b)

The Final Answer

Therefore, the minimum value of the function y = a * cos(bx) occurs at the value of x given by:

x = π/2(1 - 1/b)

This is the final answer to the problem.

Conclusion

In this article, we discussed the minimum value of the function y = a * cos(bx) and determined the value of x at which it occurs. We used the properties of the cosine function and the given value of a to find the minimum value of the function. The final answer is x = π/2(1 - 1/b).

Introduction

In our previous article, we discussed the minimum value of the function y = a * cos(bx) and determined the value of x at which it occurs. In this article, we will answer some frequently asked questions about the minimum value of the function.

**Q: What is the minimum value of the function y = a * cos(bx)?**

A: The minimum value of the function y = a * cos(bx) is -a.

Q: At what value of x does the minimum value of the function occur?

A: The minimum value of the function occurs at the value of x given by:

x = π/2(1 - 1/b)

Q: What is the relationship between the amplitude a and the frequency b?

A: The amplitude a and the frequency b are related by the equation:

a = 1300* (10 + 35/b)

Q: How do I find the value of n that satisfies the inequality n < 2b - 1/2?

A: To find the value of n, you can use the inequality:

n < 2b - 1/2

You can then find the smallest integer value of n that satisfies this inequality.

Q: What is the significance of the value of x = π/2(1 - 1/b)?

A: The value of x = π/2(1 - 1/b) is the value of x at which the minimum value of the function y = a * cos(bx) occurs.

Q: Can I use the value of x = π/2(1 - 1/b) to find the minimum value of the function?

A: Yes, you can use the value of x = π/2(1 - 1/b) to find the minimum value of the function. Simply substitute this value of x into the equation for the function:

y = a * cos(bx)

Q: What is the relationship between the minimum value of the function and the amplitude a?

A: The minimum value of the function is equal to the negative of the amplitude a:

y = -a

Q: Can I use the value of x = π/2(1 - 1/b) to find the value of the amplitude a?

A: Yes, you can use the value of x = π/2(1 - 1/b) to find the value of the amplitude a. Simply substitute this value of x into the equation for the function:

y = a * cos(bx)

And then use the fact that the minimum value of the function is equal to the negative of the amplitude a:

y = -a

**Q: What is the significance of the value of b in the equation a = 1300* (10 + 35/b)?**

A: The value of b in the equation a = 1300* (10 + 35/b) determines the frequency of the function.

Q: Can I use the value of b to find the value of the amplitude a?

A: Yes, you can use the value of b to find the value of the amplitude a. Simply substitute the value of b into the equation:

a = 1300* (10 + 35/b)

Conclusion

In this article, we answered some frequently asked questions about the minimum value of the function y = a * cos(bx). We discussed the relationship between the amplitude a and the frequency b, and how to use the value of x = π/2(1 - 1/b) to find the minimum value of the function. We also discussed the significance of the value of b in the equation a = 1300* (10 + 35/b).