Th range of k for which the inequaliity `kcos^2x-kcosx+1>=0 AA x in(-oo,oo) is`

Options

A. `klt- 1/2`
B. `kgt4`
C. `-1/2lekle4`
D. `1/2lekle5`

Explanation

To find the range of \( k \) for which the inequality

\[ k \cos^2 x - k \cos x + 1 \geq 0 \]

holds for all \( x \in (-\infty, \infty) \), we can follow these steps:

### Step 1: Rewrite the Inequality
We start with the inequality:

\[ k \cos^2 x - k \cos x + 1 \geq 0 \]

This can be treated as a quadratic in terms of \( \cos x \):

\[ k y^2 - k y + 1 \geq 0 \]

where \( y = \cos x \).

### Step 2: Determine the Discriminant
For the quadratic \( k y^2 - k y + 1 \) to be non-negative for all values of \( y \), its discriminant must be less than or equal to zero. The discriminant \( D \) is given by:

\[ D = b^2 - 4ac = (-k)^2 - 4(k)(1) = k^2 - 4k \]

We require:

\[ D \leq 0 \]

### Step 3: Solve the Discriminant Inequality
Setting the discriminant less than or equal to zero gives:

\[ k^2 - 4k \leq 0 \]

Factoring the left-hand side:

\[ k(k - 4) \leq 0 \]

### Step 4: Find the Critical Points
The critical points of the inequality are \( k = 0 \) and \( k = 4 \). We can analyze the sign of the product \( k(k - 4) \) in the intervals determined by these points:

1. For \( k < 0 \): \( k(k - 4) > 0 \)
2. For \( 0 < k < 4 \): \( k(k - 4) < 0 \)
3. For \( k > 4 \): \( k(k - 4) > 0 \)

### Step 5: Determine the Valid Range
The inequality \( k(k - 4) \leq 0 \) holds true in the interval:

\[ 0 \leq k \leq 4 \]

### Step 6: Include the Endpoints
Since the inequality is non-strict (i.e., \( \geq 0 \)), we include the endpoints:

\[ k \in [0, 4] \]

### Conclusion
Thus, the range of \( k \) for which the inequality \( k \cos^2 x - k \cos x + 1 \geq 0 \) holds for all \( x \) is:

\[ \boxed{[0, 4]} \]