Optimizing the Maximum Value of the Function fx sin(x)cos(x) for SEO

Understanding the maximum value of a function is crucial in various fields, including mathematics, engineering, and physics. This article delves into the process of finding the maximum value of the function fx sin(x)cos(x) using both trigonometric identities and differential calculus. By following these steps, the article aims to assist SEO experts in optimizing content related to trigonometric functions and optimization techniques.

Introduction to the Function

The function fx sin(x)cos(x) is a product of two trigonometric functions. To find its maximum value, we can utilize the double angle identity to simplify the function and then apply calculus to find the critical points.

Step 1: Use the Double Angle Identity

The double angle identity for sine is given by sin(2x) 2 × sin(x) × cos(x). Using this identity, we can rewrite the function as:

fx sin(x)cos(x) frac{1}{2}sin(2x)

Step 2: Analyze the Function

To find the maximum value of sin(2x), we need to identify where the sine function reaches its maximum value. The sine function has a maximum value of 1, which occurs at 2x frac{pi}{2} 2kpi for any integer k. Hence, the maximum value of fx can be calculated as:

fx frac{1}{2} × 1 frac{1}{2}

Step 3: Conclusion

Therefore, the maximum value of the function fx sin(x)cos(x) is frac{1}{2}.

boxed{frac{1}{2}}

Different Perspectives on Finding the Maximum Value

Simplified Approach Using Trigonometric Identities

We can also express the function as fx frac{sin(2x)}{2}. Given that the maximum value of sin(2x) is 1, substituting this value gives:

fx frac{1}{2}

Calculus Approach to Find Maxima

To find the critical points of the function fx sin(x)cos(x), we first find its derivative:

f'(x) cos^2(x) - sin^2(x) cos(2x)

Setting the derivative to zero to find critical points:

cos(2x) 0

Which implies:

2x frac{pi}{2} kpi, where k is an integer.

solving for x:

x frac{pi}{4} kfrac{pi}{2}

Evaluating the second derivative f''(x) -2sin(2x) at these points:

When x frac{pi}{4}, f''(x) -1, indicating a maximum.

The maximum value of the function fx is:

fleft(frac{pi}{4}right) sinleft(frac{pi}{4}right)cosleft(frac{pi}{4}right) left(frac{1}{sqrt{2}}right)left(frac{1}{sqrt{2}}right) frac{1}{2}

Generalized Approach Using Trigonometric Maximum Value

We know that the sine function ranges from -1 to 1. Therefore:

-1 ≤ sin(2x) ≤ 1

Multiplying by 1/2:

-frac{1}{2} ≤ frac{sin(2x)}{2} ≤ frac{1}{2}

This confirms that the maximum value of fx is frac{1}{2}.

Conclusion

By using both trigonometric identities and calculus, we have successfully determined that the maximum value of the function fx sin(x)cos(x) is frac{1}{2}. This approach can be applied to similar functions and is valuable for SEO in explaining and optimizing for content on mathematical optimization and trigonometry.