Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x) = {\log _{\sqrt m }}\{ \sqrt 2 (\sin x - \cos x) + m - 2\}$, for some $m$, such that the range of $f$ is [0, 2]. Then the value of $m$ is _________
Solution
We know that $\sin x-\cos x \in[-\sqrt{2}, \sqrt{2}]$
<br/><br/>
$$
\begin{aligned}
& \log _{\sqrt{M}}(\sqrt{2}(\sin x-\cos ) +M-2) \\\\
&\quad\quad\in {\left[\log _{\sqrt{M}}(M-4), \log _{\sqrt{M}} M\right] }
\end{aligned}
$$
<br/><br/>
$\Rightarrow \log _{\sqrt{M}}(M-4)=0 \Rightarrow M=5$
About this question
Subject: Mathematics · Chapter: Sets, Relations and Functions · Topic: Sets and Operations
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