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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions1 Mark
Question
Function $\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$ is monotonic decreasing when:
  1. $\lambda>\frac{1}{2}$
  2. $\lambda<\frac{1}{2}$
  3. $\lambda<2$
  4. $\lambda>2$
✓

Answer

  1. $\lambda>\frac{1}{2}$

Solution:

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

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

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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