Question:
The number of elements in the set
\[
\{ x \in \mathbb{R} : 8x^2 + x^4 + x^8 = \cos x \}
\]
is equal to:
The number of elements in the set
\[
\{ x \in \mathbb{R} : 8x^2 + x^4 + x^8 = \cos x \}
\]
is equal to:
Show Hint
When solving equations involving polynomials and trigonometric functions, analyze the growth behavior of both sides. If one side grows without bound and the other is bounded, there may be a finite number of intersections.
Updated On: Sep 6, 2025
- 0
- 1
- 2
- greater than or equal to 3
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The Correct Option is C
Solution and Explanation
Step 1: Analyze the equation.
We are given the equation \( 8x^2 + x^4 + x^8 = \cos x \), and we need to find the number of solutions to this equation. Step 2: Investigate the behavior of both sides.
The left-hand side of the equation is a polynomial function \( 8x^2 + x^4 + x^8 \), which is a strictly increasing function for all \( x \). As \( |x| \) increases, the left-hand side grows rapidly. The right-hand side, \( \cos x \), oscillates between -1 and 1. Step 3: Check for possible intersections.
Given that the polynomial on the left-hand side increases without bound while the cosine function is bounded, there will be two points of intersection where the two sides are equal. This results in exactly two solutions for the equation. Final Answer: \[ \boxed{2}. \]
We are given the equation \( 8x^2 + x^4 + x^8 = \cos x \), and we need to find the number of solutions to this equation. Step 2: Investigate the behavior of both sides.
The left-hand side of the equation is a polynomial function \( 8x^2 + x^4 + x^8 \), which is a strictly increasing function for all \( x \). As \( |x| \) increases, the left-hand side grows rapidly. The right-hand side, \( \cos x \), oscillates between -1 and 1. Step 3: Check for possible intersections.
Given that the polynomial on the left-hand side increases without bound while the cosine function is bounded, there will be two points of intersection where the two sides are equal. This results in exactly two solutions for the equation. Final Answer: \[ \boxed{2}. \]
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