Application of DerivativeHard
Question
For the function f(x) = x cot-1x, x ≥ 0
Options
A.there is atleast one x ∈ (0, 1) for which cot-1x = 
B.for atleast one x in the internal (0, ∞), f
-f(x) < 1
-f(x) < 1C.number of solution of the equation f(x) = sec x is 1
D.f′(x) is strictly decreasing is the internal (0, ∞)
Solution
f(0) = 0 ≠ f(1)
there will be no x ∈ (0, ∈) (∴ Rolle′s theorem is not applicable)
for which f′(x) = 0 i.e, cot-1 x =
f′(x) ∝
f″(x) =
f(0+) =
c ∈ 
(∴ lmvt is applicable)
f′(x) ≥ 0; f(x) is increasing
∴ f(x) ∈ [f(0), f(∞))
f(0) = 0
=
f(x) ∈ [0, 1)
f(x) = sec x will have no solution
there will be no x ∈ (0, ∈) (∴ Rolle′s theorem is not applicable)
for which f′(x) = 0 i.e, cot-1 x =
f′(x) ∝
f″(x) =
f(0+) =
c ∈ (∴ lmvt is applicable)f′(x) ≥ 0; f(x) is increasing
∴ f(x) ∈ [f(0), f(∞))
f(0) = 0
=
f(x) ∈ [0, 1)
f(x) = sec x will have no solution
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