First slide

Functions (XII)

Question

$If f (x) = \frac{x^{2} + 1}{[x]}$ , ([⋅] denotes the greatest integer function), 1 ≤ x < 4, then

Moderate

A

range of f is 2 $[2, \frac{17}{3})$
B

f is monotonically increasing in [1, 4]
C

the maximum value of f (x) is 17/3
D

the maximum value of f (x) is 17/4

Solution

We have, $f (x) = \frac{x^{2} + 1}{[x]}$
When x ∈ [1, 2) then f (x) = x² + 1 ⇒ R_f = [2, 5).
When x ∈ [2, 3) then $f (x) = \frac{x^{2} + 1}{2} \Rightarrow R_{f} = [\frac{5}{2}, 5)$ .
When x ∈ [3, 4) then $f (x) = \frac{x^{2} + 1}{3} \Rightarrow R_{f} = [\frac{10}{3}, \frac{17}{3})$ .
$∴$ R_f = [2, 17/3).

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