Question

Let P(x)=cos²x+sin⁴x,for any x∈R.Then which of the following options is correct for all x?

• A. \(\frac{1}{6}\leq P(x)\leq\frac{3}{4}\)
• B. \(0\leq P(x)\leq\frac{1}{2}\)
• C. 0≤P(x)≤1
• D. \(\frac{1}{2}\leq P(x)\leq\frac{3}{2}\)
• E. \(\frac{3}{4}\leq P(x)\leq1\)

Answer 1

Answer: (E)

Given:

• The function \( P(x) = \cos^2{x} + \sin^4{x} \) for any real \( x \)

Step 1: Express \( P(x) \) in terms of \( \sin^2{x} \): \[ P(x) = \cos^2{x} + \sin^4{x} = (1 - \sin^2{x}) + \sin^4{x} = 1 - \sin^2{x} + \sin^4{x} \]

Step 2: Let \( y = \sin^2{x} \) (note \( 0 \leq y \leq 1 \)): \[ P(x) = 1 - y + y^2 \]

Step 3: Analyze the quadratic function \( f(y) = y^2 - y + 1 \):

• The vertex occurs at \( y = \frac{1}{2} \)
• Minimum value: \( f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^2 - \frac{1}{2} + 1 = \frac{3}{4} \)
• Maximum value: \( f(0) = f(1) = 1 \) (at the endpoints)

 

Step 4: Determine the range of \( P(x) \): \[ \frac{3}{4} \leq P(x) \leq 1 \]

The correct option is (E) \( \frac{3}{4} \leq P(x) \leq 1 \).

Answer 2

Let \( P(x) = \cos^2(x) + \sin^4(x) \). We want to find the range of \( P(x) \) for all \( x \in \mathbb{R} \).

We can rewrite \( P(x) \) as:

\[ P(x) = \cos^2(x) + \sin^4(x) = \cos^2(x) + \sin^2(x)\sin^2(x) = \cos^2(x) + (1 - \cos^2(x))(1 - \cos^2(x)) \] \[ = \cos^2(x) + 1 - 2\cos^2(x) + \cos^4(x) = \cos^4(x) - \cos^2(x) + 1 \]

Let \( u = \cos^2(x) \). Since \( 0 \leq \cos^2(x) \leq 1 \), we have \( 0 \leq u \leq 1 \). Then:

\[ P(x) = u^2 - u + 1 \]

This is a quadratic function in \( u \). To find its minimum and maximum values on the interval \([0, 1]\), we can complete the square:

\[ P(x) = (u - \frac{1}{2})^2 - \frac{1}{4} + 1 = (u - \frac{1}{2})^2 + \frac{3}{4} \]

• Minimum: The minimum occurs at \( u = \frac{1}{2} \) (the vertex of the parabola), and the minimum value is \( \frac{3}{4} \).
• Maximum: The maximum occurs at \( u = 0 \) or \( u = 1 \) (endpoints of the interval). 
• In both cases, \( P(x) = 1 \).

Therefore, the range of \( P(x) \) is \([ \frac{3}{4}, 1 ]\).

So the correct option is: \( \frac{3}{4} \leq P(x) \leq 1 \).