	The difference between the maximum and minimum value of the function f(x)=3sin^4 x-cos^6 x is:
A. 3/2		B. 5/2
C. 3		D. 4


To find the difference between the maximum and minimum value of the function \( f(x) = 3\sin^4 x - \cos^6 x \), we can analyze the function step by step.

### Step 1: Rewrite the function in terms of a single trigonometric function.
We know that \( \cos^2 x = 1 - \sin^2 x \). So, \( \cos^6 x = (1 - \sin^2 x)^3 \).

Thus, the function becomes:
\[
f(x) = 3\sin^4 x - (1 - \sin^2 x)^3
\]

### Step 2: Analyze the possible values of \( \sin^2 x \).
The function \( \sin^2 x \) ranges between 0 and 1. Therefore, we substitute \( y = \sin^2 x \), where \( y \in [0, 1] \).

The function now becomes:
\[
f(y) = 3y^2 - (1 - y)^3
\]

### Step 3: Simplify and find critical points.
Expanding \( (1 - y)^3 \), we get:
\[
(1 - y)^3 = 1 - 3y + 3y^2 - y^3
\]
Thus,
\[
f(y) = 3y^2 - [1 - 3y + 3y^2 - y^3] = 3y^2 - 1 + 3y - 3y^2 + y^3
\]
Simplifying further:
\[
f(y) = y^3 + 3y - 1
\]

### Step 4: Calculate the maximum and minimum values.
To find the maximum and minimum values of \( f(y) \) in the interval \( [0, 1] \), we need to evaluate \( f(y) \) at the boundaries and check the derivative to find any local extrema.

1. **At \( y = 0 \)**:
   \[
   f(0) = 0^3 + 3(0) - 1 = -1
   \]

2. **At \( y = 1 \)**:
   \[
   f(1) = 1^3 + 3(1) - 1 = 3
   \]

Thus, the function has a minimum value of -1 and a maximum value of 3.

### Step 5: Find the difference between the maximum and minimum values.
\[
\text{Difference} = 3 - (-1) = 4
\]

### Final Answer:
The difference between the maximum and minimum values of the function is **4**.

Thus, the correct answer is **D. 4**.

---

Would you like more details or have any questions? Here are 8 related questions:

1. How do you find the derivative of \( f(x) = 3\sin^4 x - \cos^6 x \)?
2. What are the steps to expand \( (1 - y)^3 \)?
3. How do you find the critical points of a cubic function?
4. What is the range of \( \sin^2 x \) and \( \cos^2 x \) for trigonometric functions?
5. How do you determine if a function has a maximum or minimum at a critical point?
6. Can you explain how to evaluate a function at boundary points?
7. How would the problem change if the function were \( f(x) = \sin^4 x - \cos^6 x \)?
8. How do you find the maximum and minimum values of a trigonometric function?

**Tip:** When solving trigonometric functions, it’s helpful to express the function in terms of a single trigonometric identity, as this can simplify calculations significantly.

Explore how to find the difference between the maximum and minimum values of the function f(x)=3sin^4 x - cos^6 x. Detailed step-by-step solution including trigonometric identities and function analysis. Suitable for high school students studying trigonometry.

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