If-f-x-2cos-2-x-2-1-3-cos-x-then-find-the-range-of-R-f-

Question Number 182332 by mnjuly1970 last updated on 07/Dec/22

If, f (x) = 2 {c o s}^{2} (\frac{x}{2}) - ⌊ \frac{1}{3} + c o s (x) ⌋

t h e n f i n d t h e r a n g e o f : R_{f}

Answered by floor(10²Eta[1]) last updated on 07/Dec/22

- 1 ⩽ cosx ⩽ 1 \Rightarrow \frac{- 2}{3} ⩽ \frac{1}{3} + cosx ⩽ \frac{4}{3}

\Rightarrow - 1 ⩽ ⌊ \frac{1}{3} + cosx ⌋ ⩽ 1 \Rightarrow - 1 ⩽ - ⌊ \frac{1}{3} + cosx ⌋ ⩽ 1

0 ⩽ {2 \cos}^{2} (\frac{x}{2}) ⩽ 2

\Rightarrow - 1 ⩽ f (x) ⩽ 3

Answered by mnjuly1970 last updated on 08/Dec/22

f (x) = 1 + c o s (x) - ⌊ \frac{1}{3} + c o s (x) ⌋

= \frac{1}{3} + c o s (x) - ⌊ \frac{1}{3_{}} + c o s (x) ⌋ + \frac{2}{3}

\forall u \in R, 0 ⩽ u - ⌊ u ⌋ < 1 \Rightarrow f (x) \in [\frac{2}{3}, \frac{5}{3})