Function f(x)= -2 is monotonic decreasing when : - Vidyadip

MCQ

Function $\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$ is monotonic decreasing when :

A
$\lambda > \frac{1}{2}$
B
$\lambda < \frac{1}{2}$
✓
$\lambda < 2$
D
$\lambda > 2$

✓

Answer

Correct option: C.

$\lambda < 2$

$\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$
$\text{f}'(\text{x})=-\sin\text{x}-2\lambda$
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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