If `cos^(4) x + sin^(2) x -lamda =0, lamda in R` has real solutions, then

Options

A. `lamda le 1`
B. `(3)/(4) le lamda le 1`
C. `lamda ge (3)/(4)`
D. none

Answer

To solve the equation \( \cos^4 x + \sin^2 x - \lambda = 0 \) for real solutions in terms of \( \lambda \), we can follow these steps:

### Step 1: Rewrite \( \cos^4 x \)

We start with the equation:
\[ \cos^4 x + \sin^2 x - \lambda = 0 \]
Using the identity \( \cos^2 x = 1 - \sin^2 x \), we can express \( \cos^4 x \) as:
\[ \cos^4 x = (\cos^2 x)^2 = (1 - \sin^2 x)^2 \]
So we rewrite the equation:
\[ (1 - \sin^2 x)^2 + \sin^2 x - \lambda = 0 \]

### Step 2: Expand the equation

Expanding \( (1 - \sin^2 x)^2 \):
\[ (1 - \sin^2 x)^2 = 1 - 2\sin^2 x + \sin^4 x \]
Substituting this back into the equation gives:
\[ 1 - 2\sin^2 x + \sin^4 x + \sin^2 x - \lambda = 0 \]
Combining like terms results in:
\[ \sin^4 x - \sin^2 x + (1 - \lambda) = 0 \]

### Step 3: Let \( y = \sin^2 x \)

Let \( y = \sin^2 x \). The equation becomes:
\[ y^2 - y + (1 - \lambda) = 0 \]

### Step 4: Determine the discriminant

For this quadratic equation in \( y \) to have real solutions, the discriminant must be non-negative:
\[ D = b^2 - 4ac = (-1)^2 - 4(1)(1 - \lambda) \geq 0 \]
This simplifies to:
\[ 1 - 4(1 - \lambda) \geq 0 \]
\[ 1 - 4 + 4\lambda \geq 0 \]
\[ 4\lambda - 3 \geq 0 \]
\[ \lambda \geq \frac{3}{4} \]

### Step 5: Find the upper limit for \( \lambda \)

Since \( y = \sin^2 x \) must be in the range \([0, 1]\), we also need to ensure that the maximum value of \( y \) does not exceed 1. The maximum value of the quadratic \( y^2 - y + (1 - \lambda) \) occurs at:
\[ y = \frac{-b}{2a} = \frac{1}{2} \]
Substituting \( y = \frac{1}{2} \) into the equation:
\[ \left(\frac{1}{2}\right)^2 - \frac{1}{2} + (1 - \lambda) = 0 \]
This gives:
\[ \frac{1}{4} - \frac{1}{2} + 1 - \lambda = 0 \]
\[ \frac{1}{4} - \frac{2}{4} + 1 - \lambda = 0 \]
\[ \frac{3}{4} - \lambda = 0 \]
Thus,
\[ \lambda = \frac{3}{4} \]

### Conclusion

Combining both conditions, we find that for the equation to have real solutions, \( \lambda \) must satisfy:
\[ \frac{3}{4} \leq \lambda \leq 1 \]