1. What is minimum & maximum value of sin²x+cos⁴x. how can it be calculated.

To find the minimum and maximum values of the expression sin²(x) + cos⁴(x), we can use a combination of trigonometric identities and calculus. Let's break it down step by step.

Understanding the Components

The expression consists of two parts: sin²(x) and cos⁴(x). We know that:

• sin²(x) ranges from 0 to 1, since the sine function oscillates between -1 and 1.
• cos(x) also ranges from -1 to 1, so cos²(x) similarly ranges from 0 to 1, and thereby cos⁴(x) will also range from 0 to 1.

Expressing in Terms of One Variable

To simplify the calculation, we can express cos⁴(x) in terms of sin²(x). Using the identity cos²(x) = 1 - sin²(x), we can write:

cos⁴(x) = (cos²(x))² = (1 - sin²(x))²

Now, we can redefine our function:

f(sin²(x)) = sin²(x) + (1 - sin²(x))²

Finding the Maximum and Minimum Values

Let’s set y = sin²(x), where y ranges from 0 to 1. The function now becomes:

f(y) = y + (1 - y)²

Expanding this gives:

f(y) = y + (1 - 2y + y²) = y² - y + 1

Calculating the Derivative

To find the extrema, we need to take the derivative of the function:

f'(y) = 2y - 1

Setting the derivative to zero to find critical points:

2y - 1 = 0

y = 0.5

Evaluating the Function at Critical Points and Endpoints

Now, we evaluate f(y) at y = 0, y = 0.5, and y = 1:

• f(0) = 0 + (1 - 0)² = 1
• f(0.5) = 0.5 + (1 - 0.5)² = 0.5 + 0.25 = 0.75
• f(1) = 1 + (1 - 1)² = 1

Determining the Maximum and Minimum Values

From our evaluations:

• The minimum value occurs at y = 0.5, giving us f(0.5) = 0.75.
• The maximum value is f(0) = 1 and f(1) = 1.

Thus, the minimum value of sin²(x) + cos⁴(x) is 0.75, and the maximum value is 1.

Final Insights

We can conclude that the expression sin²(x) + cos⁴(x) varies between 0.75 and 1, depending on the value of x. This demonstrates how trigonometric functions can interact to form bounded expressions, and it emphasizes the importance of understanding the properties of these functions in calculus.