The function `f:R to R` is defined by `f(x)=cos^(2)x+sin^(4)x` for `x in R`. Then the range of `f(x)` is

Options

A. `((3)/(4),1]`
B. `[(3)/(4),1)`
C. `[(3)/(4),1]`
D. `((3)/(4),1)`

Explanation

To find the range of the function \( f(x) = \cos^2 x + \sin^4 x \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = \cos^2 x + \sin^4 x \]
We can express \( \sin^4 x \) in terms of \( \cos^2 x \) using the identity \( \sin^2 x = 1 - \cos^2 x \):
\[ \sin^4 x = (\sin^2 x)^2 = (1 - \cos^2 x)^2 \]
Thus, we can rewrite \( f(x) \) as:
\[ f(x) = \cos^2 x + (1 - \cos^2 x)^2 \]

### Step 2: Expand the expression
Now, we expand \( (1 - \cos^2 x)^2 \):
\[ (1 - \cos^2 x)^2 = 1 - 2\cos^2 x + \cos^4 x \]
Substituting this back into \( f(x) \):
\[ f(x) = \cos^2 x + 1 - 2\cos^2 x + \cos^4 x \]
This simplifies to:
\[ f(x) = 1 - \cos^2 x + \cos^4 x \]

### Step 3: Let \( y = \cos^2 x \)
Let \( y = \cos^2 x \). Since \( \cos^2 x \) ranges from 0 to 1, we can substitute \( y \) into the function:
\[ f(y) = 1 - y + y^2 \]

### Step 4: Find the range of \( f(y) \)
Now we need to analyze the function \( f(y) = y^2 - y + 1 \) for \( y \in [0, 1] \). This is a quadratic function that opens upwards (since the coefficient of \( y^2 \) is positive).

### Step 5: Calculate the vertex
The vertex of the quadratic \( f(y) = ay^2 + by + c \) is given by \( y = -\frac{b}{2a} \):
\[ y = -\frac{-1}{2 \cdot 1} = \frac{1}{2} \]
Now we evaluate \( f(y) \) at the endpoints and the vertex:
1. At \( y = 0 \):
\[ f(0) = 0^2 - 0 + 1 = 1 \]
2. At \( y = 1 \):
\[ f(1) = 1^2 - 1 + 1 = 1 \]
3. At \( y = \frac{1}{2} \):
\[ f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \frac{1}{2} + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{3}{4} \]

### Step 6: Determine the range
Now we have:
- \( f(0) = 1 \)
- \( f(1) = 1 \)
- \( f\left(\frac{1}{2}\right) = \frac{3}{4} \)

The minimum value of \( f(y) \) is \( \frac{3}{4} \) and the maximum value is \( 1 \). Therefore, the range of \( f(x) \) is:
\[ \left[\frac{3}{4}, 1\right] \]

### Final Answer
The range of the function \( f(x) = \cos^2 x + \sin^4 x \) is:
\[ \left[\frac{3}{4}, 1\right] \]