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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions3 Marks
Question
Write the set of values of a for which $\text{f}(\text{x})=\cos\text{x}+\text{a}^2\text{x}+\text{b}$ is strictly increasing on R.
✓

Answer

$\text{f}(\text{x})=\cos\text{x}+\text{a}^2\text{x}+\text{b}$$\text{f}'(\text{x})=\text{a}^2-\sin\text{x}$
Given: f(x) is strictly increasing on R. $\Rightarrow\text{f}'(\text{x})>0,\forall\ \text{x}\in\text{R}$ $\Rightarrow\text{a}^2-\sin\text{x}>0,\forall\ \text{x}\in\text{R}$ $\Rightarrow\text{a}^2>\sin\text{x},\forall\ \text{x}\in\text{R}$ We know that the maximum value of $\sin\text{x}$ is 1. Since, $\text{a}^2>\sin\text{x},\text{a}^2$ is always greater than 1. $\Rightarrow\text{a}^2>1$ $\Rightarrow\text{a}^2-1>0$ $\Rightarrow(\text{a}+1)(\text{a}-1)>0$ $\Rightarrow\text{a}\in(-\infty,-1)\cup(1,\infty)$

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