First slide

Functions (XII)

Question

Which of the following belongs to the range set of $C o t^{- 1} (x^{2} - 4 x + 5)$ ?

Difficult

A

$\frac{2}{3}$
B

$\frac{3}{4}$
C

$\frac{5}{6}$
D

$\frac{7}{8}$

Solution

We have $0 \leq {(x - 2)}^{2} < \infty$
$\Rightarrow 1 \leq (x^{2} - 4 x + 5) < \infty$

$\Rightarrow 0 < C o t^{- 1} (x^{2} - 4 x + 5) \leq \frac{π}{4}$

$R a n g e = (0, \frac{π}{4}]$

Clearly $\frac{2}{3} < \frac{π}{4} a n d \frac{3}{4} < \frac{π}{4} a r e t r u e$

Also $\frac{5}{6} < \frac{π}{4} a n d \frac{7}{8} < \frac{π}{4} a r e f a l s e$

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