To determine the minimum value of the expression [tex]\(1 + 2 \cos (4x)\)[/tex] for all real numbers [tex]\(x\)[/tex], we need to closely examine the behavior of the cosine function.
### Step-by-Step Solution:
1. Understand the Range of the Cosine Function:
- The cosine function [tex]\(\cos(4x)\)[/tex] oscillates between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex]. That is, for any real number [tex]\(x\)[/tex], [tex]\(-1 \leq \cos(4x) \leq 1\)[/tex].
2. Determine the Expression for Extremes:
- Given the expression [tex]\(1 + 2 \cos(4x)\)[/tex], we need to see how it behaves at the extreme values of [tex]\(\cos(4x)\)[/tex]. Consider what happens when [tex]\(\cos(4x)\)[/tex] achieves its minimum and maximum values.
3. Evaluate at the Minimum Value of [tex]\(\cos(4x)\)[/tex]:
- The minimum value of [tex]\(\cos(4x)\)[/tex] is [tex]\(-1\)[/tex].
- Substituting [tex]\(-1\)[/tex] into the expression:
[tex]\[
1 + 2 \cos(4x) \rightarrow 1 + 2(-1) = 1 - 2 = -1
\][/tex]
4. Evaluate at the Maximum Value of [tex]\(\cos(4x)\)[/tex]:
- The maximum value of [tex]\(\cos(4x)\)[/tex] is [tex]\(1\)[/tex].
- Substituting [tex]\(1\)[/tex] into the expression:
[tex]\[
1 + 2 \cos(4x) \rightarrow 1 + 2(1) = 1 + 2 = 3
\][/tex]
5. Conclusion:
- The value of [tex]\(1 + 2 \cos(4x)\)[/tex] ranges from [tex]\(-1\)[/tex] to [tex]\(3\)[/tex].
- Therefore, the minimum value of the expression [tex]\(1 + 2 \cos(4x)\)[/tex] is [tex]\(-1\)[/tex].
Hence, for all real numbers [tex]\(x\)[/tex], the minimum value of [tex]\(1 + 2 \cos(4x)\)[/tex] is [tex]\(-1\)[/tex].
