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 \(\cos^2 x\):
\[ \sin^4 x = (\sin^2 x)^2 = (1 - \cos^2 x)^2 \]
So, we 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 our function gives:
\[ f(x) = \cos^2 x + 1 - 2\cos^2 x + \cos^4 x \]
This simplifies to:
\[ f(x) = 1 - \cos^2 x + \cos^4 x \]

### Step 3: Substitute \(y = \cos^2 x\)
Let \(y = \cos^2 x\). Since \(\cos^2 x\) can take values in the range \([0, 1]\), we substitute \(y\) into the function:
\[ f(y) = 1 - y + y^2 \]

### Step 4: Analyze the function \(f(y)\)
We need to find the range of the quadratic function:
\[ f(y) = y^2 - y + 1 \]
This is a parabola that opens upwards (since the coefficient of \(y^2\) is positive).

### Step 5: Find the vertex
The vertex of a quadratic function \(ax^2 + bx + c\) occurs at \(y = -\frac{b}{2a}\). Here, \(a = 1\) and \(b = -1\):
\[ y = -\frac{-1}{2 \cdot 1} = \frac{1}{2} \]
Now we evaluate \(f\) at \(y = \frac{1}{2}\):
\[ f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right) + 1 = \frac{1}{4} - \frac{1}{2} + 1 = \frac{1}{4} - \frac{2}{4} + \frac{4}{4} = \frac{3}{4} \]

### Step 6: Evaluate endpoints
Now we evaluate \(f(y)\) at the endpoints \(y = 0\) and \(y = 1\):
- For \(y = 0\):
\[ f(0) = 0^2 - 0 + 1 = 1 \]
- For \(y = 1\):
\[ f(1) = 1^2 - 1 + 1 = 1 \]

### Step 7: Determine the range
The minimum value of \(f(y)\) occurs at the vertex \(y = \frac{1}{2}\) and is \(\frac{3}{4}\). The maximum value occurs at the endpoints \(y = 0\) and \(y = 1\) and is \(1\).

Thus, the range of \(f(x)\) is:
\[ \boxed{\left[\frac{3}{4}, 1\right]} \]