Let f : R R be a function defined by f ( x ) = log m { 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 [2023]

Question

Let $f : R \to R$ 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 [2023]

Smart Achievers · Accepted Answer

(2)

Since, $- \sqrt{2} \leq \sin x - \cos x \leq \sqrt{2}$

$∴$ $- 2 \leq \sqrt{2} (\sin x - \cos x) \leq 2$

Let $f$

$- 2 \leq t \leq 2$ ...(i)

$f (x) = \log_{\sqrt{m}} (t + m - 2)$

We have, $0 \leq f (x) \leq 2$

$\Rightarrow 0 \leq \log_{\sqrt{m}} (t + m - 2) \leq 2$

$1 \leq t + m - 2 \leq m$

$- m + 3 \leq t \leq 2$ ...(ii)

From (i) and (ii), we get $- m + 3 = - 2 \Rightarrow m = 5$

Solution

Q.
Let $f : R \to R$ 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 [2023]

1 3

2 5

3 4

4 2

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Solution

Q. Let f:R→R be a function defined by f(x)=logm{2(sinx-cosx)+m-2}, for some m, such that the range of f is [0, 2]. Then the value of m is [2023]

1 3

2 5

3 4

4 2

Ans. (2) Since, -2≤sinx-cosx≤2 ∴ -2≤2(sinx-cosx)≤2 Let 2(sinx-cosx)=t -2≤t≤2 ...(i) f(x)=logm(t+m-2) We have, 0≤f(x)≤2 ⇒ 0≤logm(t+m-2)≤2 1≤t+m-2≤m -m+3≤t≤2 ...(ii) From (i) and (ii), we get -m+3=-2 ⇒ m=5

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Q.
Let $f : R \to R$ 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 [2023]