Question
Function $\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$ is monotonic decreasing when:
- $\lambda>\frac{1}{2}$
- $\lambda<\frac{1}{2}$
- $\lambda<2$
- $\lambda>2$
✓
Answer
- $\lambda>\frac{1}{2}$
$\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$
$\text{f}'(\text{x})=-\sin\text{x}-2\lambda$
For f(x) to be decreasing, we must have
$\text{f}'(\text{x})<0$
$\Rightarrow-\sin\text{x}-2\lambda<0$
$\Rightarrow\sin\text{x}+2\lambda>0$
$\Rightarrow2\lambda>-\sin\text{x}$
We know that the maximum value of $-\sin\text{x}$ is 1.
$\Rightarrow2\lambda>1$
$\Rightarrow\lambda>\frac{1}{2}$
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