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CGP EDU Academic Team
Published on: August 12, 2026
The function f (x) = 2 log (x – 2) – x 2 + 4x + 1 increases in the interval.
Text Solution
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CHECK THE SOLUTION.
(b,c)
f ′ (x) = 
– 2x + 4
= 
– 2 (x – 2) = 2 
Therefore f ′ (x) > 0 only if (i) 1 – (x – 2) 2 > 0 and x – 2 > 0, or (ii) 1 – (x – 2) 2 < 0 and
x – 2 < 0. In the first case, we have (x – 3) (x – 1) < 0 and x > 2. That is, 1 < x < 3 and x > 2, so that
x ∈ (2, 3). The second case is not possible as the domain of f(x) is {x : x > 2}.
Hence f (x) increases on (2, 3) and
since (5/2, 3) ⊂ (2, 3), it increase on (5/2, 3) as well.B
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