Let $\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by $\mathrm{f}(\mathrm{x})= \log _{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}$, for some $m$, such that the range of f is $[0,2]$. Then the value of m is

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Sets Relations and Functions

Let

f : R \to R

be a function defined by

f (x) = lo g_{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

Answer: (A)

Writing

k = 2 (sin x - cos x) = 2 sin (x - \frac{π}{4})

, the quantity

k

ranges over

[- 2, 2]

, so the argument

k + m - 2

ranges over

[m - 4, m]

. For a base

m > 1

the logarithm increases, so its greatest value is

lo g_{m} m = 2

, which already matches the top of the required range. The least value must be

0

, so

m - 4 = 1

, giving

m = 5

JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Single correct option