Let [f:RR] be a function defined by [f(x) = _ m 2 ( x - x) + m - 2 ], for some m, such tha…

Question

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 \_\_\_\_\_\_\_\_\_

Accepted Answer

We know that $\sin x-\cos x \in[-\sqrt{2}, \sqrt{2}]$

$$
\begin{aligned}
& \log _{\sqrt{M}}(\sqrt{2}(\sin x-\cos ) +M-2) \\
&\quad\quad\in {\left[\log _{\sqrt{M}}(M-4), \log _{\sqrt{M}} M\right] }
\end{aligned}
$$

$\Rightarrow \log _{\sqrt{M}}(M-4)=0 \Rightarrow M=5$

Solutions · 1