Mathematics Question on Functions

Question

Let $f: R \rightarrow R$ 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_____

Answer 1

Solution

The correct answer is (B) : 5
Since,
−2≤sinx−cosx≤2
∴−2≤2(sinx−cosx)≤2
( Assume 2(sinx−cosx)=k)
−2≤k≤2… (i)
f(x)=logm(k+m−2)
Given,
0≤f(x)≤2
0≤logm(k+m−2)≤2
1≤k+m−2≤m
−m+3≤k≤2… (ii)
From eq. (i) & (ii), we get −m+3=−2
⇒m=5