Find which of the two is larger n (1 + x) or 
.
Text Solution
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n (1 + x)
Sol. Let x > –1
Consider f(x) = (1 + x) n(1 + x) – tan –1 x
f ′ (x) = n(1 + x) + 1 – 
f ′′ (x) = 
> 0 ⇒ f ′ (x) is increasing
For x < 0, f ′ (x) < f ′ (0) ⇒ f ′ (x) < 0 ⇒ f(x) is decreasing ⇒ f(x) > f(0) ⇒ f(x) > 0
⇒ (1 + x) n(1 + x) – tan –1 x > 0
n(1 + x) > 
For x > 0, f ′ (x) > f ′ (0) ⇒ f ′ (x) > 0 ⇒ f(x) is increasing ⇒ f(x) > f(0)
⇒ f(x) > 0 ⇒ n(1 + x) > 
Hence larger of these is n(1 + x).
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